Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.2% → 45.3%

Time: 9.7s

Alternatives: 18

Speedup: 0.7×

• could not determine a ground truth

  1. c0 = 1.9455374766095425e+307
  2. w = 2.2349286819203452e-297
  3. h = -8.88082367942193e-256
  4. D = -2.5631104248459926e-248
  5. d = 3.977101034190539e+300
  6. M = -1.6844992530996406e-295

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.2% 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: 45.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.2%

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

   3. Applied rewrites 27.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 27.0

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

   5. Applied rewrites 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 34.7

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

   7. Applied rewrites 34.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 34.7

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

   9. Applied rewrites 34.7%

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

   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 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 2: 45.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

   2. Taylor expanded in D around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

   8. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 14.4

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

   10. Applied rewrites 14.4%

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

   11. Taylor expanded in w around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 17.7

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

   13. Applied rewrites 17.7%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(D \cdot D\right) \cdot w}, \sqrt{\left(c0 \cdot c0\right) \cdot \frac{{d}^{4}}{{D}^{4} \cdot \left(w \cdot w\right)}}\right) \cdot c0}{h \cdot w} \cdot 0.5} \]

   5. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 13.8%

      \[\leadsto \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{w \cdot w}} + \frac{c0 \cdot \left(d \cdot d\right)}{w}\right)}{D \cdot D}}{h \cdot w} \cdot 0.5 \]

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 18.4

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

   10. Applied rewrites 18.4%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 3: 44.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.2%

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

   3. Applied rewrites 27.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 27.0

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

   5. Applied rewrites 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 34.7

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

   7. Applied rewrites 34.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 34.3

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

   9. Applied rewrites 34.3%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 34.7

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

   11. Applied rewrites 34.7%

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

   13. Applied rewrites 32.7%

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

   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 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 4: 44.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.2%

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

   3. Applied rewrites 27.8%

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

   5. Applied rewrites 31.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-+.f64 32.6

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

   7. Applied rewrites 32.6%

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

   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 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 5: 41.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.2%

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

   3. Applied rewrites 27.8%

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

   5. Applied rewrites 31.3%

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

   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 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 6: 39.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

   2. Taylor expanded in D around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in d around 0

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

   10. Applied rewrites 22.8%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(D \cdot D\right) \cdot w}, \sqrt{\left(c0 \cdot c0\right) \cdot \frac{{d}^{4}}{{D}^{4} \cdot \left(w \cdot w\right)}}\right) \cdot c0}{h \cdot w} \cdot 0.5} \]

   5. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 13.8%

      \[\leadsto \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{w \cdot w}} + \frac{c0 \cdot \left(d \cdot d\right)}{w}\right)}{D \cdot D}}{h \cdot w} \cdot 0.5 \]

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 18.4

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

   10. Applied rewrites 18.4%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 7: 37.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

   2. Taylor expanded in D around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in w around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-flip N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 17.1

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

   10. Applied rewrites 17.1%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(D \cdot D\right) \cdot w}, \sqrt{\left(c0 \cdot c0\right) \cdot \frac{{d}^{4}}{{D}^{4} \cdot \left(w \cdot w\right)}}\right) \cdot c0}{h \cdot w} \cdot 0.5} \]

   5. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 13.8%

      \[\leadsto \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{w \cdot w}} + \frac{c0 \cdot \left(d \cdot d\right)}{w}\right)}{D \cdot D}}{h \cdot w} \cdot 0.5 \]

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 18.4

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

   10. Applied rewrites 18.4%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 8: 37.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

   2. Taylor expanded in D around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in w around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 11.7%

      \[\leadsto \left(c0 \cdot \frac{-\frac{\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{h \cdot h}} + \left(-\frac{c0 \cdot \left(d \cdot d\right)}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   8. Taylor expanded in d around 0

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

      1. sub-flip N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-flip N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 16.8

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

   10. Applied rewrites 16.8%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(D \cdot D\right) \cdot w}, \sqrt{\left(c0 \cdot c0\right) \cdot \frac{{d}^{4}}{{D}^{4} \cdot \left(w \cdot w\right)}}\right) \cdot c0}{h \cdot w} \cdot 0.5} \]

   5. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 13.8%

      \[\leadsto \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{w \cdot w}} + \frac{c0 \cdot \left(d \cdot d\right)}{w}\right)}{D \cdot D}}{h \cdot w} \cdot 0.5 \]

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 18.4

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

   10. Applied rewrites 18.4%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 9: 36.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

   2. Taylor expanded in D around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 17.2

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

   10. Applied rewrites 17.2%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(D \cdot D\right) \cdot w}, \sqrt{\left(c0 \cdot c0\right) \cdot \frac{{d}^{4}}{{D}^{4} \cdot \left(w \cdot w\right)}}\right) \cdot c0}{h \cdot w} \cdot 0.5} \]

   5. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 13.8%

      \[\leadsto \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{w \cdot w}} + \frac{c0 \cdot \left(d \cdot d\right)}{w}\right)}{D \cdot D}}{h \cdot w} \cdot 0.5 \]

   8. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 18.4

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

   10. Applied rewrites 18.4%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 10: 36.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

   2. Taylor expanded in D around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 17.2

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

   10. Applied rewrites 17.2%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(D \cdot D\right) \cdot w}, \sqrt{\left(c0 \cdot c0\right) \cdot \frac{{d}^{4}}{{D}^{4} \cdot \left(w \cdot w\right)}}\right) \cdot c0}{h \cdot w} \cdot 0.5} \]

   5. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 13.8%

      \[\leadsto \frac{\frac{c0 \cdot \left(\sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{w \cdot w}} + \frac{c0 \cdot \left(d \cdot d\right)}{w}\right)}{D \cdot D}}{h \cdot w} \cdot 0.5 \]

   8. Taylor expanded in d around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

   10. Applied rewrites 18.4%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 11: 36.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 25.2%

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

   2. Taylor expanded in D around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 17.2

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

   10. Applied rewrites 17.2%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in D around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 18.4

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

   10. Applied rewrites 18.4%

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

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

   1. Initial program 25.2%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      10. lower-neg.f64 22.2

          \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.2%

      \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 36.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(D \cdot D\right) \cdot w\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := \frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{t\_0}\right) \cdot 0.5\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{w \cdot w}} + \frac{c0}{w}\right)}{h}}{t\_0}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* D D) w))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 (- INFINITY))
     (*
      (*
       c0
       (/ (/ (* (* d d) (+ (sqrt (/ (* c0 c0) (* h h))) (/ c0 h))) w) t_0))
      0.5)
     (if (<= t_2 INFINITY)
       (*
        (*
         c0
         (/ (/ (* (* d d) (+ (sqrt (/ (* c0 c0) (* w w))) (/ c0 w))) h) t_0))
        0.5)
       (* (/ (* (pow (* (- M) M) 0.5) c0) w) 0.5)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * D) * w;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (c0 * ((((d * d) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / w) / t_0)) * 0.5;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (c0 * ((((d * d) * (sqrt(((c0 * c0) / (w * w))) + (c0 / w))) / h) / t_0)) * 0.5;
	} else {
		tmp = ((pow((-M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * D) * w;
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = (c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (c0 * ((((d * d) * (Math.sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / w) / t_0)) * 0.5;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * ((((d * d) * (Math.sqrt(((c0 * c0) / (w * w))) + (c0 / w))) / h) / t_0)) * 0.5;
	} else {
		tmp = ((Math.pow((-M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (D * D) * w
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = (c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (c0 * ((((d * d) * (math.sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / w) / t_0)) * 0.5
	elif t_2 <= math.inf:
		tmp = (c0 * ((((d * d) * (math.sqrt(((c0 * c0) / (w * w))) + (c0 / w))) / h) / t_0)) * 0.5
	else:
		tmp = ((math.pow((-M * M), 0.5) * c0) / w) * 0.5
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(D * D) * w)
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(c0 * Float64(Float64(Float64(Float64(d * d) * Float64(sqrt(Float64(Float64(c0 * c0) / Float64(h * h))) + Float64(c0 / h))) / w) / t_0)) * 0.5);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(c0 * Float64(Float64(Float64(Float64(d * d) * Float64(sqrt(Float64(Float64(c0 * c0) / Float64(w * w))) + Float64(c0 / w))) / h) / t_0)) * 0.5);
	else
		tmp = Float64(Float64(Float64((Float64(Float64(-M) * M) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (D * D) * w;
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = (c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (c0 * ((((d * d) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / w) / t_0)) * 0.5;
	elseif (t_2 <= Inf)
		tmp = (c0 * ((((d * d) * (sqrt(((c0 * c0) / (w * w))) + (c0 / w))) / h) / t_0)) * 0.5;
	else
		tmp = ((((-M * M) ^ 0.5) * c0) / w) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(c0 * N[(N[(N[(N[(d * d), $MachinePrecision] * N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(c0 * N[(N[(N[(N[(d * d), $MachinePrecision] * N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -inf.0

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in D around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w} \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w} \cdot \color{blue}{\frac{1}{2}} \]

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

      \[\leadsto \left(c0 \cdot \frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      2. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      4. lower-+.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      9. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      10. pow2 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      15. lift-*.f64 22.8

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   7. Applied rewrites 22.8%

      \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   8. Taylor expanded in w around 0

      \[\leadsto \left(c0 \cdot \frac{\frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      3. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      10. pow2 N/A

          \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      12. lower-/.f64 17.2

          \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   10. Applied rewrites 17.2%

       \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   if -inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in D around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w} \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w} \cdot \color{blue}{\frac{1}{2}} \]

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

      \[\leadsto \left(c0 \cdot \frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      2. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      4. lower-+.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      9. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      10. pow2 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      15. lift-*.f64 22.8

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   7. Applied rewrites 22.8%

      \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   8. Taylor expanded in h around 0

      \[\leadsto \left(c0 \cdot \frac{\frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      3. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{w}^{2}}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      10. pow2 N/A

          \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{w \cdot w}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{w \cdot w}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      12. lower-/.f64 18.6

          \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{w \cdot w}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   10. Applied rewrites 18.6%

       \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{w \cdot w}} + \frac{c0}{w}\right)}{h}}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.3

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. pow1/2 N/A

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      10. lower-neg.f64 22.2

          \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.2%

      \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 31.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (*
      (*
       c0
       (/
        (/ (* (* d d) (+ (sqrt (/ (* c0 c0) (* h h))) (/ c0 h))) w)
        (* (* D D) w)))
      0.5)
     (* (/ (* (pow (* (- M) M) 0.5) c0) w) 0.5))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (c0 * ((((d * d) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / w) / ((D * D) * w))) * 0.5;
	} else {
		tmp = ((pow((-M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * ((((d * d) * (Math.sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / w) / ((D * D) * w))) * 0.5;
	} else {
		tmp = ((Math.pow((-M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 * ((((d * d) * (math.sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / w) / ((D * D) * w))) * 0.5
	else:
		tmp = ((math.pow((-M * M), 0.5) * c0) / w) * 0.5
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 * Float64(Float64(Float64(Float64(d * d) * Float64(sqrt(Float64(Float64(c0 * c0) / Float64(h * h))) + Float64(c0 / h))) / w) / Float64(Float64(D * D) * w))) * 0.5);
	else
		tmp = Float64(Float64(Float64((Float64(Float64(-M) * M) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * ((((d * d) * (sqrt(((c0 * c0) / (h * h))) + (c0 / h))) / w) / ((D * D) * w))) * 0.5;
	else
		tmp = ((((-M * M) ^ 0.5) * c0) / w) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * N[(N[(N[(N[(d * d), $MachinePrecision] * N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in D around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w} \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \left(\sqrt{\frac{{c0}^{2} \cdot {d}^{4}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w} \cdot \color{blue}{\frac{1}{2}} \]

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(c0 \cdot \frac{\mathsf{fma}\left(c0, \frac{d \cdot d}{h \cdot w}, \sqrt{\frac{\left(c0 \cdot c0\right) \cdot {d}^{4}}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5} \]

   5. Taylor expanded in d around 0

      \[\leadsto \left(c0 \cdot \frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      2. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      4. lower-+.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2} \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      9. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot {w}^{2}}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      10. pow2 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      14. lower-/.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      15. lift-*.f64 22.8

          \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   7. Applied rewrites 22.8%

      \[\leadsto \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot h\right) \cdot \left(w \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   8. Taylor expanded in w around 0

      \[\leadsto \left(c0 \cdot \frac{\frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{{d}^{2} \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      3. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{{c0}^{2}}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{{h}^{2}}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      10. pow2 N/A

          \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot \frac{1}{2} \]

      12. lower-/.f64 17.2

          \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   10. Applied rewrites 17.2%

       \[\leadsto \left(c0 \cdot \frac{\frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{h \cdot h}} + \frac{c0}{h}\right)}{w}}{\left(D \cdot D\right) \cdot w}\right) \cdot 0.5 \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.3

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. pow1/2 N/A

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      10. lower-neg.f64 22.2

          \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.2%

      \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 28.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + \sqrt{-M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (+ t_1 (sqrt (- (* M M)))))
     (* (/ (* (pow (* (- M) M) 0.5) c0) w) 0.5))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_1 + sqrt(-(M * M)));
	} else {
		tmp = ((pow((-M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_1 + Math.sqrt(-(M * M)));
	} else {
		tmp = ((Math.pow((-M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * (t_1 + math.sqrt(-(M * M)))
	else:
		tmp = ((math.pow((-M * M), 0.5) * c0) / w) * 0.5
	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(-Float64(M * M)))));
	else
		tmp = Float64(Float64(Float64((Float64(Float64(-M) * M) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * (t_1 + sqrt(-(M * M)));
	else
		tmp = ((((-M * M) ^ 0.5) * c0) / w) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$1 + N[Sqrt[(-N[(M * M), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 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. mul-1-neg N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\mathsf{neg}\left({M}^{2}\right)}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{-{M}^{2}}\right) \]

      3. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{-M \cdot M}\right) \]

      4. lift-*.f64 8.4

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{-M \cdot M}\right) \]

   4. Applied rewrites 8.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\color{blue}{-M \cdot M}}\right) \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.3

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. pow1/2 N/A

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      10. lower-neg.f64 22.2

          \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.2%

      \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 28.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, \sqrt{-M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (fma c0 (/ (* d d) (* (* (* h w) D) D)) (sqrt (- (* M M)))))
     (* (/ (* (pow (* (- M) M) 0.5) c0) w) 0.5))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * fma(c0, ((d * d) / (((h * w) * D) * D)), sqrt(-(M * M)));
	} else {
		tmp = ((pow((-M * M), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * fma(c0, Float64(Float64(d * d) / Float64(Float64(Float64(h * w) * D) * D)), sqrt(Float64(-Float64(M * M)))));
	else
		tmp = Float64(Float64(Float64((Float64(Float64(-M) * M) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(c0 * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] + N[Sqrt[(-N[(M * M), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left({M}^{2}\right)} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \color{blue}{\sqrt{\mathsf{neg}\left({M}^{2}\right)}}\right) \]

      2. associate-/l* N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(c0 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \sqrt{\color{blue}{\mathsf{neg}\left({M}^{2}\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(c0, \color{blue}{\frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}, \sqrt{\mathsf{neg}\left({M}^{2}\right)}\right) \]

   4. Applied rewrites 9.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, \sqrt{-M \cdot M}\right)} \]

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 25.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.3

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. pow1/2 N/A

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      10. lower-neg.f64 22.2

          \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.2%

      \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 22.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ (* (pow (* (- M) M) 0.5) c0) w) 0.5))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return ((pow((-M * M), 0.5) * c0) / w) * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = ((((-m * m) ** 0.5d0) * c0) / w) * 0.5d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((Math.pow((-M * M), 0.5) * c0) / w) * 0.5;
}

Python

def code(c0, w, h, D, d, M):
	return ((math.pow((-M * M), 0.5) * c0) / w) * 0.5

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64((Float64(Float64(-M) * M) ^ 0.5) * c0) / w) * 0.5)
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = ((((-M * M) ^ 0.5) * c0) / w) * 0.5;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 25.2%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

   7. lower-neg.f64 N/A

      \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   8. pow2 N/A

      \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

   9. lift-*.f64 14.3

      \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

4. Applied rewrites 14.3%

   \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

   2. pow1/2 N/A

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{{\left(-M \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   5. pow2 N/A

      \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   7. pow2 N/A

      \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   10. lower-neg.f64 22.2

       \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

6. Applied rewrites 22.2%

   \[\leadsto \frac{{\left(\left(-M\right) \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]
7. Add Preprocessing

Alternative 17: 14.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ (* (sqrt (- (* M M))) c0) w) 0.5))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return ((sqrt(-(M * M)) * c0) / w) * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = ((sqrt(-(m * m)) * c0) / w) * 0.5d0
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) * 0.5;
}

Python

def code(c0, w, h, D, d, M):
	return ((math.sqrt(-(M * M)) * c0) / w) * 0.5

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(sqrt(Float64(-Float64(M * M))) * c0) / w) * 0.5)
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = ((sqrt(-(M * M)) * c0) / w) * 0.5;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(N[Sqrt[(-N[(M * M), $MachinePrecision])], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 25.2%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \color{blue}{\frac{1}{2}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \cdot \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{\sqrt{\mathsf{neg}\left({M}^{2}\right)} \cdot c0}{w} \cdot \frac{1}{2} \]

   7. lower-neg.f64 N/A

      \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   8. pow2 N/A

      \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

   9. lift-*.f64 14.3

      \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

4. Applied rewrites 14.3%

   \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
5. Add Preprocessing

Alternative 18: 0.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{w + w} \cdot \left(\sqrt{-1} \cdot M\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ c0 (+ w w)) (* (sqrt -1.0) M)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (w + w)) * (sqrt(-1.0) * 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
    code = (c0 / (w + w)) * (sqrt((-1.0d0)) * m)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (w + w)) * (Math.sqrt(-1.0) * M);
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (w + w)) * (math.sqrt(-1.0) * M)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(w + w)) * Float64(sqrt(-1.0) * M))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (w + w)) * (sqrt(-1.0) * M);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[-1.0], $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.2%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

2. Taylor expanded in M around inf

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot \color{blue}{M}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot \color{blue}{M}\right) \]

   3. lower-sqrt.f64 0.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot M\right) \]

4. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

   2. count-2-rev N/A

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

   3. lower-+.f64 0.0

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

6. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\frac{c0}{w + w} \cdot \left(\sqrt{-1} \cdot M\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(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))))))