ABCF->ab-angle b

Percentage Accurate: 18.6% → 48.0%

Time: 13.8s

Alternatives: 14

Speedup: 16.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Initial Program: 18.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Alternative 1: 48.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{B\_m \cdot B\_m}{C}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := {B\_m}^{2} - t\_1\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := 2 \cdot \left(\left(B\_m \cdot B\_m - t\_1\right) \cdot F\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{\left(-B\_m\right) \cdot B\_m + t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_4 \cdot \left(\left(A + -0.5 \cdot t\_0\right) + A\right)}}{-C \cdot \left(t\_0 - 4 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (* B_m B_m) C))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- (pow B_m 2.0) t_1))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (* 2.0 (* (- (* B_m B_m) t_1) F))))
   (if (<= t_3 (- INFINITY))
     (-
      (sqrt
       (*
        (/
         (* F (- (+ A C) (hypot B_m (- A C))))
         (- (* B_m B_m) (* 4.0 (* A C))))
        2.0)))
     (if (<= t_3 -2e-209)
       (/
        (sqrt (* t_4 (- (+ A C) (hypot (- A C) B_m))))
        (+ (* (- B_m) B_m) t_1))
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* t_4 (+ (+ A (* -0.5 t_0)) A)))
          (- (* C (- t_0 (* 4.0 A)))))
         (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) / C;
	double t_1 = (4.0 * A) * C;
	double t_2 = pow(B_m, 2.0) - t_1;
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = 2.0 * (((B_m * B_m) - t_1) * F);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -sqrt((((F * ((A + C) - hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_3 <= -2e-209) {
		tmp = sqrt((t_4 * ((A + C) - hypot((A - C), B_m)))) / ((-B_m * B_m) + t_1);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_4 * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)));
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) / C;
	double t_1 = (4.0 * A) * C;
	double t_2 = Math.pow(B_m, 2.0) - t_1;
	double t_3 = Math.sqrt(((2.0 * (t_2 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_2;
	double t_4 = 2.0 * (((B_m * B_m) - t_1) * F);
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -Math.sqrt((((F * ((A + C) - Math.hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_3 <= -2e-209) {
		tmp = Math.sqrt((t_4 * ((A + C) - Math.hypot((A - C), B_m)))) / ((-B_m * B_m) + t_1);
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_4 * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)));
	} else {
		tmp = -Math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (B_m * B_m) / C
	t_1 = (4.0 * A) * C
	t_2 = math.pow(B_m, 2.0) - t_1
	t_3 = math.sqrt(((2.0 * (t_2 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_2
	t_4 = 2.0 * (((B_m * B_m) - t_1) * F)
	tmp = 0
	if t_3 <= -math.inf:
		tmp = -math.sqrt((((F * ((A + C) - math.hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0))
	elif t_3 <= -2e-209:
		tmp = math.sqrt((t_4 * ((A + C) - math.hypot((A - C), B_m)))) / ((-B_m * B_m) + t_1)
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_4 * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)))
	else:
		tmp = -math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(B_m * B_m) / C)
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64((B_m ^ 2.0) - t_1)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_1) * F))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(A + C) - hypot(B_m, Float64(A - C)))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C)))) * 2.0)));
	elseif (t_3 <= -2e-209)
		tmp = Float64(sqrt(Float64(t_4 * Float64(Float64(A + C) - hypot(Float64(A - C), B_m)))) / Float64(Float64(Float64(-B_m) * B_m) + t_1));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_4 * Float64(Float64(A + Float64(-0.5 * t_0)) + A))) / Float64(-Float64(C * Float64(t_0 - Float64(4.0 * A)))));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m * B_m) / C;
	t_1 = (4.0 * A) * C;
	t_2 = (B_m ^ 2.0) - t_1;
	t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_2;
	t_4 = 2.0 * (((B_m * B_m) - t_1) * F);
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -sqrt((((F * ((A + C) - hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	elseif (t_3 <= -2e-209)
		tmp = sqrt((t_4 * ((A + C) - hypot((A - C), B_m)))) / ((-B_m * B_m) + t_1);
	elseif (t_3 <= Inf)
		tmp = sqrt((t_4 * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)));
	else
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], (-N[Sqrt[N[(N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$3, -2e-209], N[(N[Sqrt[N[(t$95$4 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$4 * N[(N[(A + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(t$95$0 - N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000001e-209

   1. Initial program 98.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]

   if -2.0000000000000001e-209 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 15.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 40.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 40.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{C \cdot \left(\frac{{B}^{2}}{C} - 4 \cdot A\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \color{blue}{\left(\frac{{B}^{2}}{C} - 4 \cdot A\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{{B}^{2}}{C} - \color{blue}{4 \cdot A}\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - \color{blue}{4} \cdot A\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      6. lift-*.f64 40.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot \color{blue}{A}\right)} \]

   8. Applied rewrites 40.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      3. lift-*.f64 40.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

   10. Applied rewrites 40.8%

       \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)}} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 16.4

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 16.4%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 20.9

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 20.9%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) + A\right)}}{-C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := {B\_m}^{2} - t\_0\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-154}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\right) \cdot \frac{\mathsf{fma}\left(-0.5, B\_m \cdot B\_m, C \cdot \left(A + A\right)\right)}{C}}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- (pow B_m 2.0) t_0))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1))))
   (if (<= t_2 -2e-154)
     (-
      (sqrt
       (*
        (/
         (* F (- (+ A C) (hypot B_m (- A C))))
         (- (* B_m B_m) (* 4.0 (* A C))))
        2.0)))
     (if (<= t_2 INFINITY)
       (/
        (sqrt
         (*
          (* 2.0 (* (- (* B_m B_m) t_0) F))
          (/ (fma -0.5 (* B_m B_m) (* C (+ A A))) C)))
        (+ (* (- B_m) B_m) t_0))
       (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = pow(B_m, 2.0) - t_0;
	double t_2 = sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_1;
	double tmp;
	if (t_2 <= -2e-154) {
		tmp = -sqrt((((F * ((A + C) - hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (fma(-0.5, (B_m * B_m), (C * (A + A))) / C))) / ((-B_m * B_m) + t_0);
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64((B_m ^ 2.0) - t_0)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_1))
	tmp = 0.0
	if (t_2 <= -2e-154)
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(A + C) - hypot(B_m, Float64(A - C)))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C)))) * 2.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F)) * Float64(fma(-0.5, Float64(B_m * B_m), Float64(C * Float64(A + A))) / C))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-154], (-N[Sqrt[N[(N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-154

   1. Initial program 41.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   if -1.9999999999999999e-154 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 20.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 39.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 39.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 39.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C} \]

      5. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 39.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 39.6%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]

   8. Taylor expanded in C around 0

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\frac{-1}{2} \cdot {B}^{2} + C \cdot \left(A - -1 \cdot A\right)}{\color{blue}{C}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\frac{-1}{2} \cdot {B}^{2} + C \cdot \left(A - -1 \cdot A\right)}{C}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{2}, {B}^{2}, C \cdot \left(A - -1 \cdot A\right)\right)}{C}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      3. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{2}, B \cdot B, C \cdot \left(A - -1 \cdot A\right)\right)}{C}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{2}, B \cdot B, C \cdot \left(A - -1 \cdot A\right)\right)}{C}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{2}, B \cdot B, C \cdot \left(A - -1 \cdot A\right)\right)}{C}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{2}, B \cdot B, C \cdot \left(A - -1 \cdot A\right)\right)}{C}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      7. lift-*.f64 39.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\mathsf{fma}\left(-0.5, B \cdot B, C \cdot \left(A - -1 \cdot A\right)\right)}{C}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   10. Applied rewrites 39.6%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\mathsf{fma}\left(-0.5, B \cdot B, C \cdot \left(A - -1 \cdot A\right)\right)}{\color{blue}{C}}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 2.9%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 16.4

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 16.4%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 20.9

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 20.9%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \frac{\mathsf{fma}\left(-0.5, B \cdot B, C \cdot \left(A + A\right)\right)}{C}}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{--4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-117)
   (/
    (sqrt (* (* 2.0 (* (- (* B_m B_m) (* (* 4.0 A) C)) F)) (* 2.0 A)))
    (- (* -4.0 (* A C))))
   (if (<= (pow B_m 2.0) 5e+303)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A (sqrt (fma A A (* B_m B_m)))))))
     (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-117) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * (2.0 * A))) / -(-4.0 * (A * C));
	} else if (pow(B_m, 2.0) <= 5e+303) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m))))));
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-117)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(2.0 * A))) / Float64(-Float64(-4.0 * Float64(A * C))));
	elseif ((B_m ^ 2.0) <= 5e+303)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m)))))));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-117], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+303], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e-117

   1. Initial program 18.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 28.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 28.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 28.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C} \]

      5. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 28.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 28.3%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot A\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. lift-*.f64 26.4

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot \color{blue}{A}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   10. Applied rewrites 26.4%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot A\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   11. Taylor expanded in A around inf

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{-4 \cdot \color{blue}{\left(A \cdot C\right)}} \]

       2. lower-*.f64 25.4

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{-4 \cdot \left(A \cdot \color{blue}{C}\right)} \]

   13. Applied rewrites 25.4%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

   if 5e-117 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999997e303

   1. Initial program 35.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F} \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      7. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + {B}^{2}}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      10. lower-hypot.f64 22.6

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right) \]

   5. Applied rewrites 22.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      2. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + B \cdot B}\right)}\right) \]

      3. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      5. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + {B}^{2}}\right)}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, {B}^{2}\right)}\right)}\right) \]

      7. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\right) \]

      8. lift-*.f64 21.3

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\right) \]

   7. Applied rewrites 21.3%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\right) \]

   if 4.9999999999999997e303 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 2.4%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 26.8

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 26.8%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 34.2

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 34.2%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{--4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{\left(-B\_m\right) \cdot B\_m + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-117)
   (/
    (sqrt (* -8.0 (* A (* C (* F (+ A A))))))
    (+ (* (- B_m) B_m) (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 5e+303)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A (sqrt (fma A A (* B_m B_m)))))))
     (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-117) {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / ((-B_m * B_m) + ((4.0 * A) * C));
	} else if (pow(B_m, 2.0) <= 5e+303) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m))))));
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-117)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / Float64(Float64(Float64(-B_m) * B_m) + Float64(Float64(4.0 * A) * C)));
	elseif ((B_m ^ 2.0) <= 5e+303)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m)))))));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-117], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+303], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e-117

   1. Initial program 18.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 28.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 28.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 28.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C} \]

      5. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 28.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 28.3%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]

   8. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 25.3

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   10. Applied rewrites 25.3%

       \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   if 5e-117 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999997e303

   1. Initial program 35.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F} \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      7. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + {B}^{2}}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      10. lower-hypot.f64 22.6

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right) \]

   5. Applied rewrites 22.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. lift-hypot.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      2. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + B \cdot B}\right)}\right) \]

      3. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      5. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + {B}^{2}}\right)}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, {B}^{2}\right)}\right)}\right) \]

      7. pow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\right) \]

      8. lift-*.f64 21.3

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\right) \]

   7. Applied rewrites 21.3%

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\right) \]

   if 4.9999999999999997e303 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 2.4%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 26.8

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 26.8%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 34.2

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 34.2%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-27}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{\left(-B\_m\right) \cdot B\_m + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+71}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-27)
   (/
    (sqrt (* -8.0 (* A (* C (* F (+ A A))))))
    (+ (* (- B_m) B_m) (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 5e+71)
     (- (sqrt (* (/ F C) -1.0)))
     (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-27) {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / ((-B_m * B_m) + ((4.0 * A) * C));
	} else if (pow(B_m, 2.0) <= 5e+71) {
		tmp = -sqrt(((F / C) * -1.0));
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 1d-27) then
        tmp = sqrt(((-8.0d0) * (a * (c * (f * (a + a)))))) / ((-b_m * b_m) + ((4.0d0 * a) * c))
    else if ((b_m ** 2.0d0) <= 5d+71) then
        tmp = -sqrt(((f / c) * (-1.0d0)))
    else
        tmp = -sqrt((((f * (((a + c) / b_m) - 1.0d0)) / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-27) {
		tmp = Math.sqrt((-8.0 * (A * (C * (F * (A + A)))))) / ((-B_m * B_m) + ((4.0 * A) * C));
	} else if (Math.pow(B_m, 2.0) <= 5e+71) {
		tmp = -Math.sqrt(((F / C) * -1.0));
	} else {
		tmp = -Math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-27:
		tmp = math.sqrt((-8.0 * (A * (C * (F * (A + A)))))) / ((-B_m * B_m) + ((4.0 * A) * C))
	elif math.pow(B_m, 2.0) <= 5e+71:
		tmp = -math.sqrt(((F / C) * -1.0))
	else:
		tmp = -math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-27)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) / Float64(Float64(Float64(-B_m) * B_m) + Float64(Float64(4.0 * A) * C)));
	elseif ((B_m ^ 2.0) <= 5e+71)
		tmp = Float64(-sqrt(Float64(Float64(F / C) * -1.0)));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-27)
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) / ((-B_m * B_m) + ((4.0 * A) * C));
	elseif ((B_m ^ 2.0) <= 5e+71)
		tmp = -sqrt(((F / C) * -1.0));
	else
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-27], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+71], (-N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e-27

   1. Initial program 22.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 27.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 27.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 27.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C} \]

      5. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 27.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 27.6%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]

   8. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 25.0

         \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   10. Applied rewrites 25.0%

       \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   if 1e-27 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999972e71

   1. Initial program 31.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \left(\sqrt{\frac{-1}{2}} \cdot \sqrt{2}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{C}} \cdot \left(\sqrt{\frac{-1}{2}} \cdot \sqrt{2}\right)\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\frac{-1}{2} \cdot 2}\right) \]

      3. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \color{blue}{\sqrt{-1}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\color{blue}{-1}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      7. lower-sqrt.f64 0.0

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \color{blue}{\sqrt{-1}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\color{blue}{-1}}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      5. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

      7. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

      8. lift-/.f64 28.5

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

   7. Applied rewrites 28.5%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{C} \cdot -1}} \]

   if 4.99999999999999972e71 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 22.1

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 22.1%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 26.4

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 26.4%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-27}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+71}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{B\_m \cdot B\_m}{C}\\ \mathbf{if}\;B\_m \leq 1.42 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot t\_0\right) + A\right)}}{-C \cdot \left(t\_0 - 4 \cdot A\right)}\\ \mathbf{elif}\;B\_m \leq 2.9 \cdot 10^{+175}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (* B_m B_m) C)))
   (if (<= B_m 1.42e+26)
     (/
      (sqrt
       (*
        (* 2.0 (* (- (* B_m B_m) (* (* 4.0 A) C)) F))
        (+ (+ A (* -0.5 t_0)) A)))
      (- (* C (- t_0 (* 4.0 A)))))
     (if (<= B_m 2.9e+175)
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A (hypot A B_m)))))
       (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) / C;
	double tmp;
	if (B_m <= 1.42e+26) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)));
	} else if (B_m <= 2.9e+175) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) / C;
	double tmp;
	if (B_m <= 1.42e+26) {
		tmp = Math.sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)));
	} else if (B_m <= 2.9e+175) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (A - Math.hypot(A, B_m))));
	} else {
		tmp = -Math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (B_m * B_m) / C
	tmp = 0
	if B_m <= 1.42e+26:
		tmp = math.sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)))
	elif B_m <= 2.9e+175:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (A - math.hypot(A, B_m))))
	else:
		tmp = -math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(B_m * B_m) / C)
	tmp = 0.0
	if (B_m <= 1.42e+26)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + Float64(-0.5 * t_0)) + A))) / Float64(-Float64(C * Float64(t_0 - Float64(4.0 * A)))));
	elseif (B_m <= 2.9e+175)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - hypot(A, B_m)))));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m * B_m) / C;
	tmp = 0.0;
	if (B_m <= 1.42e+26)
		tmp = sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)));
	elseif (B_m <= 2.9e+175)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	else
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]}, If[LessEqual[B$95$m, 1.42e+26], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(t$95$0 - N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.9e+175], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.42e26

   1. Initial program 22.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 21.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{C \cdot \left(\frac{{B}^{2}}{C} - 4 \cdot A\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \color{blue}{\left(\frac{{B}^{2}}{C} - 4 \cdot A\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{{B}^{2}}{C} - \color{blue}{4 \cdot A}\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - \color{blue}{4} \cdot A\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      6. lift-*.f64 21.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot \color{blue}{A}\right)} \]

   8. Applied rewrites 21.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      3. lift-*.f64 21.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

   10. Applied rewrites 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)}} \]

   if 1.42e26 < B < 2.9e175

   1. Initial program 20.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F} \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      7. lower--.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + {B}^{2}}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      10. lower-hypot.f64 46.6

          \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right) \]

   5. Applied rewrites 46.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} \]

   if 2.9e175 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 2.1%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 61.4

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 61.4%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 79.0

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 79.0%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.42 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) + A\right)}}{-C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{+175}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+71}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+71)
   (- (sqrt (* (/ F C) -1.0)))
   (- (sqrt (* -2.0 (/ F B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e+71) {
		tmp = -sqrt(((F / C) * -1.0));
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 5d+71) then
        tmp = -sqrt(((f / c) * (-1.0d0)))
    else
        tmp = -sqrt(((-2.0d0) * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e+71) {
		tmp = -Math.sqrt(((F / C) * -1.0));
	} else {
		tmp = -Math.sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e+71:
		tmp = -math.sqrt(((F / C) * -1.0))
	else:
		tmp = -math.sqrt((-2.0 * (F / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+71)
		tmp = Float64(-sqrt(Float64(Float64(F / C) * -1.0)));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e+71)
		tmp = -sqrt(((F / C) * -1.0));
	else
		tmp = -sqrt((-2.0 * (F / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+71], (-N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999972e71

   1. Initial program 24.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \left(\sqrt{\frac{-1}{2}} \cdot \sqrt{2}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{C}} \cdot \left(\sqrt{\frac{-1}{2}} \cdot \sqrt{2}\right)\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\frac{-1}{2} \cdot 2}\right) \]

      3. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \color{blue}{\sqrt{-1}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\color{blue}{-1}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      7. lower-sqrt.f64 0.0

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \color{blue}{\sqrt{-1}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\color{blue}{-1}}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      5. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

      7. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

      8. lift-/.f64 20.9

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

   7. Applied rewrites 20.9%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{C} \cdot -1}} \]

   if 4.99999999999999972e71 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 25.4%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{-2 \cdot \frac{F}{B}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{-2 \cdot \frac{F}{B}} \]

      2. lower-/.f64 25.1

         \[\leadsto -1 \cdot \sqrt{-2 \cdot \frac{F}{B}} \]

   8. Applied rewrites 25.1%

      \[\leadsto -1 \cdot \sqrt{-2 \cdot \frac{F}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+71}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{B\_m \cdot B\_m}{C}\\ \mathbf{if}\;B\_m \leq 4.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot t\_0\right) + A\right)}}{-C \cdot \left(t\_0 - 4 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (* B_m B_m) C)))
   (if (<= B_m 4.1e+36)
     (/
      (sqrt
       (*
        (* 2.0 (* (- (* B_m B_m) (* (* 4.0 A) C)) F))
        (+ (+ A (* -0.5 t_0)) A)))
      (- (* C (- t_0 (* 4.0 A)))))
     (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) / C;
	double tmp;
	if (B_m <= 4.1e+36) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)));
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b_m * b_m) / c
    if (b_m <= 4.1d+36) then
        tmp = sqrt(((2.0d0 * (((b_m * b_m) - ((4.0d0 * a) * c)) * f)) * ((a + ((-0.5d0) * t_0)) + a))) / -(c * (t_0 - (4.0d0 * a)))
    else
        tmp = -sqrt((((f * (((a + c) / b_m) - 1.0d0)) / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) / C;
	double tmp;
	if (B_m <= 4.1e+36) {
		tmp = Math.sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)));
	} else {
		tmp = -Math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (B_m * B_m) / C
	tmp = 0
	if B_m <= 4.1e+36:
		tmp = math.sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)))
	else:
		tmp = -math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(B_m * B_m) / C)
	tmp = 0.0
	if (B_m <= 4.1e+36)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + Float64(-0.5 * t_0)) + A))) / Float64(-Float64(C * Float64(t_0 - Float64(4.0 * A)))));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m * B_m) / C;
	tmp = 0.0;
	if (B_m <= 4.1e+36)
		tmp = sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * ((A + (-0.5 * t_0)) + A))) / -(C * (t_0 - (4.0 * A)));
	else
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]}, If[LessEqual[B$95$m, 4.1e+36], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(t$95$0 - N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if B < 4.10000000000000013e36

   1. Initial program 23.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 21.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{C \cdot \left(\frac{{B}^{2}}{C} - 4 \cdot A\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \color{blue}{\left(\frac{{B}^{2}}{C} - 4 \cdot A\right)}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{{B}^{2}}{C} - \color{blue}{4 \cdot A}\right)} \]

      3. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - \color{blue}{4} \cdot A\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      6. lift-*.f64 21.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot \color{blue}{A}\right)} \]

   8. Applied rewrites 21.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

      3. lift-*.f64 21.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)} \]

   10. Applied rewrites 21.6%

       \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)}} \]

   if 4.10000000000000013e36 < B

   1. Initial program 7.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 23.8%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 51.4

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 51.4%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 60.9

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 60.9%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) + A\right)}}{-C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 4.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, 2 \cdot A\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 4.1e+36)
     (/
      (sqrt
       (*
        (* 2.0 (* (- (* B_m B_m) t_0) F))
        (fma -0.5 (/ (* B_m B_m) C) (* 2.0 A))))
      (+ (* (- B_m) B_m) t_0))
     (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 4.1e+36) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * fma(-0.5, ((B_m * B_m) / C), (2.0 * A)))) / ((-B_m * B_m) + t_0);
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 4.1e+36)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F)) * fma(-0.5, Float64(Float64(B_m * B_m) / C), Float64(2.0 * A)))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 4.1e+36], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if B < 4.10000000000000013e36

   1. Initial program 23.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 21.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 21.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C} \]

      5. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 21.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 21.6%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{2 \cdot A}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{B}^{2}}{\color{blue}{C}}, 2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{C}, 2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{C}, 2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{B \cdot B}{C}, 2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

      5. lift-*.f64 21.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, 2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   10. Applied rewrites 21.6%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{B \cdot B}{C}}, 2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   if 4.10000000000000013e36 < B

   1. Initial program 7.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 23.8%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 51.4

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 51.4%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 60.9

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 60.9%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, 2 \cdot A\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.36 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(C \cdot \left(\frac{B\_m \cdot B\_m}{C} - 4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(-B\_m\right) \cdot B\_m + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.36e+35)
   (/
    (sqrt (* (* 2.0 (* (* C (- (/ (* B_m B_m) C) (* 4.0 A))) F)) (* 2.0 A)))
    (+ (* (- B_m) B_m) (* (* 4.0 A) C)))
   (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.36e+35) {
		tmp = sqrt(((2.0 * ((C * (((B_m * B_m) / C) - (4.0 * A))) * F)) * (2.0 * A))) / ((-B_m * B_m) + ((4.0 * A) * C));
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.36d+35) then
        tmp = sqrt(((2.0d0 * ((c * (((b_m * b_m) / c) - (4.0d0 * a))) * f)) * (2.0d0 * a))) / ((-b_m * b_m) + ((4.0d0 * a) * c))
    else
        tmp = -sqrt((((f * (((a + c) / b_m) - 1.0d0)) / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.36e+35) {
		tmp = Math.sqrt(((2.0 * ((C * (((B_m * B_m) / C) - (4.0 * A))) * F)) * (2.0 * A))) / ((-B_m * B_m) + ((4.0 * A) * C));
	} else {
		tmp = -Math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.36e+35:
		tmp = math.sqrt(((2.0 * ((C * (((B_m * B_m) / C) - (4.0 * A))) * F)) * (2.0 * A))) / ((-B_m * B_m) + ((4.0 * A) * C))
	else:
		tmp = -math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.36e+35)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(C * Float64(Float64(Float64(B_m * B_m) / C) - Float64(4.0 * A))) * F)) * Float64(2.0 * A))) / Float64(Float64(Float64(-B_m) * B_m) + Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.36e+35)
		tmp = sqrt(((2.0 * ((C * (((B_m * B_m) / C) - (4.0 * A))) * F)) * (2.0 * A))) / ((-B_m * B_m) + ((4.0 * A) * C));
	else
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.36e+35], N[(N[Sqrt[N[(N[(2.0 * N[(N[(C * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] - N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 1.36e35

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 21.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 21.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C} \]

      5. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 21.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 21.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot A\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. lift-*.f64 20.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot \color{blue}{A}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   10. Applied rewrites 20.3%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot A\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   11. Taylor expanded in C around inf

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\color{blue}{\left(C \cdot \left(\frac{{B}^{2}}{C} - 4 \cdot A\right)\right)} \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(C \cdot \color{blue}{\left(\frac{{B}^{2}}{C} - 4 \cdot A\right)}\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

       2. lower--.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(C \cdot \left(\frac{{B}^{2}}{C} - \color{blue}{4 \cdot A}\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

       3. pow2 N/A

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

       4. lift-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(C \cdot \left(\frac{B \cdot B}{C} - \color{blue}{4} \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

       6. lift-*.f64 20.2

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot \color{blue}{A}\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   13. Applied rewrites 20.2%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\color{blue}{\left(C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)\right)} \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   if 1.36e35 < B

   1. Initial program 7.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 23.3%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 50.1

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 50.1%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 59.5

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 59.5%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.36 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(C \cdot \left(\frac{B \cdot B}{C} - 4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.36 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.36e+35)
   (/
    (sqrt (* (* 2.0 (* (- (* B_m B_m) (* (* 4.0 A) C)) F)) (* 2.0 A)))
    (- (fma -4.0 (* A C) (* B_m B_m))))
   (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.36e+35) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - ((4.0 * A) * C)) * F)) * (2.0 * A))) / -fma(-4.0, (A * C), (B_m * B_m));
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.36e+35)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(2.0 * A))) / Float64(-fma(-4.0, Float64(A * C), Float64(B_m * B_m))));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.36e+35], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 1.36e35

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1 \cdot A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - \color{blue}{-1} \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 21.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot \color{blue}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 21.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{{B}^{2}} - \left(4 \cdot A\right) \cdot C} \]

      5. pow2 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + \frac{-1}{2} \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 21.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 21.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) - -1 \cdot A\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot A\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. lift-*.f64 20.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot \color{blue}{A}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   10. Applied rewrites 20.3%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot A\right)}}}{B \cdot B - \left(4 \cdot A\right) \cdot C} \]

   11. Taylor expanded in A around 0

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, {B}^{2}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(-4, A \cdot \color{blue}{C}, {B}^{2}\right)} \]

       3. pow2 N/A

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

       4. lift-*.f64 20.3

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   13. Applied rewrites 20.3%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   if 1.36e35 < B

   1. Initial program 7.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 23.3%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 50.1

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 50.1%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 59.5

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 59.5%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.36 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.9 \cdot 10^{+36}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B\_m} - 1\right)}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8.9e+36)
   (- (sqrt (* (/ F C) -1.0)))
   (- (sqrt (* (/ (* F (- (/ (+ A C) B_m) 1.0)) B_m) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.9e+36) {
		tmp = -sqrt(((F / C) * -1.0));
	} else {
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 8.9d+36) then
        tmp = -sqrt(((f / c) * (-1.0d0)))
    else
        tmp = -sqrt((((f * (((a + c) / b_m) - 1.0d0)) / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.9e+36) {
		tmp = -Math.sqrt(((F / C) * -1.0));
	} else {
		tmp = -Math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 8.9e+36:
		tmp = -math.sqrt(((F / C) * -1.0))
	else:
		tmp = -math.sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 8.9e+36)
		tmp = Float64(-sqrt(Float64(Float64(F / C) * -1.0)));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(Float64(A + C) / B_m) - 1.0)) / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 8.9e+36)
		tmp = -sqrt(((F / C) * -1.0));
	else
		tmp = -sqrt((((F * (((A + C) / B_m) - 1.0)) / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 8.9e+36], (-N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(N[(N[(F * N[(N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 8.89999999999999998e36

   1. Initial program 23.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \left(\sqrt{\frac{-1}{2}} \cdot \sqrt{2}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F}{C}} \cdot \left(\sqrt{\frac{-1}{2}} \cdot \sqrt{2}\right)\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\frac{-1}{2} \cdot 2}\right) \]

      3. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \color{blue}{\sqrt{-1}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\color{blue}{-1}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      7. lower-sqrt.f64 0.0

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \color{blue}{\sqrt{-1}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\color{blue}{-1}}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{-1}\right) \]

      5. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

      7. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

      8. lift-/.f64 17.1

         \[\leadsto -1 \cdot \sqrt{\frac{F}{C} \cdot -1} \]

   7. Applied rewrites 17.1%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{C} \cdot -1}} \]

   if 8.89999999999999998e36 < B

   1. Initial program 7.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   5. Applied rewrites 23.8%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

   6. Taylor expanded in B around inf

      \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{-1 \cdot F + \frac{F \cdot \left(A + C\right)}{B}}{B} \cdot 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

      5. lift-+.f64 51.4

         \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   8. Applied rewrites 51.4%

      \[\leadsto -1 \cdot \sqrt{\frac{\mathsf{fma}\left(-1, F, \frac{F \cdot \left(A + C\right)}{B}\right)}{B} \cdot 2} \]

   9. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       2. lower--.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(\frac{A}{B} + \frac{C}{B}\right) - 1\right)}{B} \cdot 2} \]

       3. div-add-rev N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

       5. lift-+.f64 60.9

          \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]

   11. Applied rewrites 60.9%

       \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.9 \cdot 10^{+36}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\frac{A + C}{B} - 1\right)}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 8.8% accurate, 15.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{A \cdot F} \cdot \frac{-2}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (sqrt (* A F)) (/ -2.0 B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((A * F)) * (-2.0 / B_m);
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((a * f)) * ((-2.0d0) / b_m)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((A * F)) * (-2.0 / B_m);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((A * F)) * (-2.0 / B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((A * F)) * (-2.0 / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.5%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   4. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F} \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   7. lower--.f64 N/A

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + {B}^{2}}\right)}\right) \]

   9. unpow2 N/A

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

   10. lower-hypot.f64 14.7

       \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right) \]

5. Applied rewrites 14.7%

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} \]

6. Taylor expanded in A around -inf

   \[\leadsto \sqrt{A \cdot F} \cdot \color{blue}{\frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{\color{blue}{B}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \]

   3. lower-*.f64 N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \]

   4. sqrt-pow2 N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{{-1}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \]

   5. metadata-eval N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{{-1}^{1} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot {\left(\sqrt{2}\right)}^{2}}{B} \]

   7. sqrt-pow2 N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot {2}^{\left(\frac{2}{2}\right)}}{B} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot {2}^{1}}{B} \]

   9. metadata-eval N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot 2}{B} \]

   10. metadata-eval N/A

       \[\leadsto \sqrt{A \cdot F} \cdot \frac{-2}{B} \]

   11. lower-/.f64 2.8

       \[\leadsto \sqrt{A \cdot F} \cdot \frac{-2}{B} \]

8. Applied rewrites 2.8%

   \[\leadsto \sqrt{A \cdot F} \cdot \color{blue}{\frac{-2}{B}} \]
9. Add Preprocessing

Alternative 14: 26.9% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{-2 \cdot \frac{F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* -2.0 (/ F B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((-2.0 * (F / B_m)));
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((-2.0d0) * (f / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((-2.0 * (F / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((-2.0 * (F / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((-2.0 * (F / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 20.5%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in F around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

   2. sqrt-unprod N/A

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

   4. lower-*.f64 N/A

      \[\leadsto -1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \cdot 2} \]

5. Applied rewrites 26.8%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}} \]

6. Taylor expanded in B around inf

   \[\leadsto -1 \cdot \sqrt{-2 \cdot \frac{F}{B}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \sqrt{-2 \cdot \frac{F}{B}} \]

   2. lower-/.f64 15.4

      \[\leadsto -1 \cdot \sqrt{-2 \cdot \frac{F}{B}} \]

8. Applied rewrites 15.4%

   \[\leadsto -1 \cdot \sqrt{-2 \cdot \frac{F}{B}} \]

9. Final simplification 15.4%

   \[\leadsto -\sqrt{-2 \cdot \frac{F}{B}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025047 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))