ABCF->ab-angle b

Percentage Accurate: 18.7% → 52.1%

Time: 11.0s

Alternatives: 9

Speedup: 20.5×

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: 18.7% 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: 52.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\\ t_2 := {B\_m}^{2} - t\_0\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \left(-B\_m\right) \cdot B\_m + t\_0\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-192}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\left(A + -0.5 \cdot \frac{B\_m \cdot B\_m}{C}\right) + A\right)}}{t\_4}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\\ \end{array} \end{array} \]

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 (* 2.0 (* (- (* B_m B_m) t_0) F)))
        (t_2 (- (pow B_m 2.0) t_0))
        (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 (+ (* (- B_m) B_m) t_0)))
   (if (<= t_3 (- INFINITY))
     (- (sqrt (* (/ F C) -1.0)))
     (if (<= t_3 -5e-192)
       (/ (sqrt (* t_1 (- (+ A C) (hypot (- A C) B_m)))) t_4)
       (if (<= t_3 5e+231)
         (/ (sqrt (* t_1 (+ (+ A (* -0.5 (/ (* B_m B_m) C))) A))) t_4)
         (if (<= t_3 INFINITY)
           (sqrt (/ (- F) C))
           (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A (hypot A 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 t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * (((B_m * B_m) - t_0) * F);
	double t_2 = pow(B_m, 2.0) - t_0;
	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 = (-B_m * B_m) + t_0;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -sqrt(((F / C) * -1.0));
	} else if (t_3 <= -5e-192) {
		tmp = sqrt((t_1 * ((A + C) - hypot((A - C), B_m)))) / t_4;
	} else if (t_3 <= 5e+231) {
		tmp = sqrt((t_1 * ((A + (-0.5 * ((B_m * B_m) / C))) + A))) / t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((-F / C));
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	}
	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 = (4.0 * A) * C;
	double t_1 = 2.0 * (((B_m * B_m) - t_0) * F);
	double t_2 = Math.pow(B_m, 2.0) - t_0;
	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 = (-B_m * B_m) + t_0;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = -Math.sqrt(((F / C) * -1.0));
	} else if (t_3 <= -5e-192) {
		tmp = Math.sqrt((t_1 * ((A + C) - Math.hypot((A - C), B_m)))) / t_4;
	} else if (t_3 <= 5e+231) {
		tmp = Math.sqrt((t_1 * ((A + (-0.5 * ((B_m * B_m) / C))) + A))) / t_4;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((-F / C));
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (A - Math.hypot(A, 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):
	t_0 = (4.0 * A) * C
	t_1 = 2.0 * (((B_m * B_m) - t_0) * F)
	t_2 = math.pow(B_m, 2.0) - t_0
	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 = (-B_m * B_m) + t_0
	tmp = 0
	if t_3 <= -math.inf:
		tmp = -math.sqrt(((F / C) * -1.0))
	elif t_3 <= -5e-192:
		tmp = math.sqrt((t_1 * ((A + C) - math.hypot((A - C), B_m)))) / t_4
	elif t_3 <= 5e+231:
		tmp = math.sqrt((t_1 * ((A + (-0.5 * ((B_m * B_m) / C))) + A))) / t_4
	elif t_3 <= math.inf:
		tmp = math.sqrt((-F / C))
	else:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (A - math.hypot(A, 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)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F))
	t_2 = Float64((B_m ^ 2.0) - t_0)
	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(Float64(Float64(-B_m) * B_m) + t_0)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-sqrt(Float64(Float64(F / C) * -1.0)));
	elseif (t_3 <= -5e-192)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) - hypot(Float64(A - C), B_m)))) / t_4);
	elseif (t_3 <= 5e+231)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(A + Float64(-0.5 * Float64(Float64(B_m * B_m) / C))) + A))) / t_4);
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(-F) / C));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - hypot(A, 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)
	t_0 = (4.0 * A) * C;
	t_1 = 2.0 * (((B_m * B_m) - t_0) * F);
	t_2 = (B_m ^ 2.0) - t_0;
	t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_2;
	t_4 = (-B_m * B_m) + t_0;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = -sqrt(((F / C) * -1.0));
	elseif (t_3 <= -5e-192)
		tmp = sqrt((t_1 * ((A + C) - hypot((A - C), B_m)))) / t_4;
	elseif (t_3 <= 5e+231)
		tmp = sqrt((t_1 * ((A + (-0.5 * ((B_m * B_m) / C))) + A))) / t_4;
	elseif (t_3 <= Inf)
		tmp = sqrt((-F / C));
	else
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, 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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $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[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], (-N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$3, -5e-192], N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 5e+231], N[(N[Sqrt[N[(t$95$1 * N[(N[(A + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], 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]]]]]]]]]]

Derivation

1. Split input into 5 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 3.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 34.6

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

   7. Applied rewrites 34.6%

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

   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))) < -5.0000000000000001e-192

   1. Initial program 98.5%

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

   4. Applied rewrites 98.5%

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

   if -5.0000000000000001e-192 < (/.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))) < 5.00000000000000028e231

   1. Initial program 30.3%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 7.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 7.3

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 7.3

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

   7. Applied rewrites 7.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 30.3

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

   10. Applied rewrites 30.3%

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

   if 5.00000000000000028e231 < (/.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 4.1%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 34.4

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

   8. Applied rewrites 34.4%

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

   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 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 19.1

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

   5. Applied rewrites 19.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-192}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) + A\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 25.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0} \leq -5 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<=
        (/
         (sqrt
          (*
           (* 2.0 (* t_0 F))
           (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
         (- t_0))
        -5e-192)
     (* (sqrt (* A F)) (/ -2.0 B_m))
     (sqrt (/ (- F) C)))))

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 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if ((sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0) <= -5e-192) {
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = sqrt((-F / C));
	}
	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 ** 2.0d0) - ((4.0d0 * a) * c)
    if ((sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b_m ** 2.0d0)))))) / -t_0) <= (-5d-192)) then
        tmp = sqrt((a * f)) * ((-2.0d0) / b_m)
    else
        tmp = sqrt((-f / c))
    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 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if ((Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_0) <= -5e-192) {
		tmp = Math.sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = Math.sqrt((-F / C));
	}
	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 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	tmp = 0
	if (math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_0) <= -5e-192:
		tmp = math.sqrt((A * F)) * (-2.0 / B_m)
	else:
		tmp = math.sqrt((-F / C))
	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((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0)) <= -5e-192)
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m));
	else
		tmp = sqrt(Float64(Float64(-F) / C));
	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 ^ 2.0) - ((4.0 * A) * C);
	tmp = 0.0;
	if ((sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_0) <= -5e-192)
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	else
		tmp = sqrt((-F / C));
	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[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], -5e-192], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 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))) < -5.0000000000000001e-192

   1. Initial program 53.0%

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

   3. Taylor expanded in C around 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 24.5

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

   5. Applied rewrites 24.5%

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

   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 4.1

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

   8. Applied rewrites 4.1%

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

   if -5.0000000000000001e-192 < (/.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 8.3%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 13.2%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 17.0

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

   8. Applied rewrites 17.0%

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

4. Final simplification 12.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{+73}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \end{array} \]

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 (<= F -2.65e+73)
   (- (sqrt (* (/ F C) -1.0)))
   (if (<= F -2e-310)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A (hypot A B_m)))))
     (sqrt (/ (- F) C)))))

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 (F <= -2.65e+73) {
		tmp = -sqrt(((F / C) * -1.0));
	} else if (F <= -2e-310) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	} else {
		tmp = sqrt((-F / C));
	}
	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 tmp;
	if (F <= -2.65e+73) {
		tmp = -Math.sqrt(((F / C) * -1.0));
	} else if (F <= -2e-310) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (A - Math.hypot(A, B_m))));
	} else {
		tmp = Math.sqrt((-F / C));
	}
	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 F <= -2.65e+73:
		tmp = -math.sqrt(((F / C) * -1.0))
	elif F <= -2e-310:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (A - math.hypot(A, B_m))))
	else:
		tmp = math.sqrt((-F / C))
	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 (F <= -2.65e+73)
		tmp = Float64(-sqrt(Float64(Float64(F / C) * -1.0)));
	elseif (F <= -2e-310)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - hypot(A, B_m)))));
	else
		tmp = sqrt(Float64(Float64(-F) / C));
	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 (F <= -2.65e+73)
		tmp = -sqrt(((F / C) * -1.0));
	elseif (F <= -2e-310)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	else
		tmp = sqrt((-F / C));
	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[F, -2.65e+73], (-N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[F, -2e-310], 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[((-F) / C), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.64999999999999998e73

   1. Initial program 13.6%

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

   3. Taylor expanded in 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 21.0

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

   7. Applied rewrites 21.0%

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

   if -2.64999999999999998e73 < F < -1.999999999999994e-310

   1. Initial program 26.6%

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

   3. Taylor expanded in C around 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 25.4

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

   5. Applied rewrites 25.4%

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

   if -1.999999999999994e-310 < F

   1. Initial program 31.9%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 38.2

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

   8. Applied rewrites 38.2%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{+73}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 2.7 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(-B\_m\right) \cdot B\_m + t\_0}\\ \mathbf{elif}\;B\_m \leq 4.5 \cdot 10^{+234}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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 2.7e-37)
     (/
      (sqrt (* (* 2.0 (* (- (* B_m B_m) t_0) F)) (* 2.0 A)))
      (+ (* (- B_m) B_m) t_0))
     (if (<= B_m 4.5e+234)
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A B_m))))
       (- (sqrt (* (/ (- F) 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 <= 2.7e-37) {
		tmp = sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (2.0 * A))) / ((-B_m * B_m) + t_0);
	} else if (B_m <= 4.5e+234) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - B_m)));
	} else {
		tmp = -sqrt(((-F / 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 = (4.0d0 * a) * c
    if (b_m <= 2.7d-37) then
        tmp = sqrt(((2.0d0 * (((b_m * b_m) - t_0) * f)) * (2.0d0 * a))) / ((-b_m * b_m) + t_0)
    else if (b_m <= 4.5d+234) then
        tmp = (sqrt(2.0d0) / -b_m) * sqrt((f * (a - b_m)))
    else
        tmp = -sqrt(((-f / 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 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 2.7e-37) {
		tmp = Math.sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (2.0 * A))) / ((-B_m * B_m) + t_0);
	} else if (B_m <= 4.5e+234) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (A - B_m)));
	} else {
		tmp = -Math.sqrt(((-F / 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 = (4.0 * A) * C
	tmp = 0
	if B_m <= 2.7e-37:
		tmp = math.sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (2.0 * A))) / ((-B_m * B_m) + t_0)
	elif B_m <= 4.5e+234:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (A - B_m)))
	else:
		tmp = -math.sqrt(((-F / 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 <= 2.7e-37)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F)) * Float64(2.0 * A))) / Float64(Float64(Float64(-B_m) * B_m) + t_0));
	elseif (B_m <= 4.5e+234)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - B_m))));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(-F) / 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 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 2.7e-37)
		tmp = sqrt(((2.0 * (((B_m * B_m) - t_0) * F)) * (2.0 * A))) / ((-B_m * B_m) + t_0);
	elseif (B_m <= 4.5e+234)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - B_m)));
	else
		tmp = -sqrt(((-F / 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[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 2.7e-37], 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[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[((-B$95$m) * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.5e+234], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[((-F) / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.70000000000000016e-37

   1. Initial program 24.4%

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

   3. Taylor expanded in B around inf

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

   5. Applied rewrites 5.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 5.1

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 5.1

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

   7. Applied rewrites 5.1%

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

   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. lower-*.f64 19.6

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

   10. Applied rewrites 19.6%

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

   if 2.70000000000000016e-37 < B < 4.49999999999999982e234

   1. Initial program 27.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 45.9

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 41.9%

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

   if 4.49999999999999982e234 < 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

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 68.4

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

   8. Applied rewrites 68.4%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.7 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(-B\right) \cdot B + \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{+234}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{-F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{+73}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \end{array} \]

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 (<= F -2.65e+73)
   (- (sqrt (* (/ F C) -1.0)))
   (if (<= F -2e-310)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A B_m))))
     (sqrt (/ (- F) C)))))

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 (F <= -2.65e+73) {
		tmp = -sqrt(((F / C) * -1.0));
	} else if (F <= -2e-310) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - B_m)));
	} else {
		tmp = sqrt((-F / C));
	}
	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 (f <= (-2.65d+73)) then
        tmp = -sqrt(((f / c) * (-1.0d0)))
    else if (f <= (-2d-310)) then
        tmp = (sqrt(2.0d0) / -b_m) * sqrt((f * (a - b_m)))
    else
        tmp = sqrt((-f / c))
    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 (F <= -2.65e+73) {
		tmp = -Math.sqrt(((F / C) * -1.0));
	} else if (F <= -2e-310) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (A - B_m)));
	} else {
		tmp = Math.sqrt((-F / C));
	}
	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 F <= -2.65e+73:
		tmp = -math.sqrt(((F / C) * -1.0))
	elif F <= -2e-310:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (A - B_m)))
	else:
		tmp = math.sqrt((-F / C))
	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 (F <= -2.65e+73)
		tmp = Float64(-sqrt(Float64(Float64(F / C) * -1.0)));
	elseif (F <= -2e-310)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - B_m))));
	else
		tmp = sqrt(Float64(Float64(-F) / C));
	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 (F <= -2.65e+73)
		tmp = -sqrt(((F / C) * -1.0));
	elseif (F <= -2e-310)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - B_m)));
	else
		tmp = sqrt((-F / C));
	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[F, -2.65e+73], (-N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[F, -2e-310], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.64999999999999998e73

   1. Initial program 13.6%

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

   3. Taylor expanded in 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 21.0

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

   7. Applied rewrites 21.0%

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

   if -2.64999999999999998e73 < F < -1.999999999999994e-310

   1. Initial program 26.6%

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

   3. Taylor expanded in C around 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 25.4

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

   5. Applied rewrites 25.4%

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

   6. Taylor expanded in A around 0

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

   8. Applied rewrites 21.6%

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

   if -1.999999999999994e-310 < F

   1. Initial program 31.9%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 38.2

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

   8. Applied rewrites 38.2%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.65 \cdot 10^{+73}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -1.25 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;-\sqrt{\frac{-F}{B\_m} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \end{array} \end{array} \]

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 (<= C -1.25e-243)
   (sqrt (/ (- F) C))
   (if (<= C 1.4e-18)
     (- (sqrt (* (/ (- F) B_m) 2.0)))
     (- (sqrt (* (/ F C) -1.0))))))

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 (C <= -1.25e-243) {
		tmp = sqrt((-F / C));
	} else if (C <= 1.4e-18) {
		tmp = -sqrt(((-F / B_m) * 2.0));
	} else {
		tmp = -sqrt(((F / C) * -1.0));
	}
	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 (c <= (-1.25d-243)) then
        tmp = sqrt((-f / c))
    else if (c <= 1.4d-18) then
        tmp = -sqrt(((-f / b_m) * 2.0d0))
    else
        tmp = -sqrt(((f / c) * (-1.0d0)))
    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 (C <= -1.25e-243) {
		tmp = Math.sqrt((-F / C));
	} else if (C <= 1.4e-18) {
		tmp = -Math.sqrt(((-F / B_m) * 2.0));
	} else {
		tmp = -Math.sqrt(((F / C) * -1.0));
	}
	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 C <= -1.25e-243:
		tmp = math.sqrt((-F / C))
	elif C <= 1.4e-18:
		tmp = -math.sqrt(((-F / B_m) * 2.0))
	else:
		tmp = -math.sqrt(((F / C) * -1.0))
	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 (C <= -1.25e-243)
		tmp = sqrt(Float64(Float64(-F) / C));
	elseif (C <= 1.4e-18)
		tmp = Float64(-sqrt(Float64(Float64(Float64(-F) / B_m) * 2.0)));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / C) * -1.0)));
	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 (C <= -1.25e-243)
		tmp = sqrt((-F / C));
	elseif (C <= 1.4e-18)
		tmp = -sqrt(((-F / B_m) * 2.0));
	else
		tmp = -sqrt(((F / C) * -1.0));
	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[C, -1.25e-243], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], If[LessEqual[C, 1.4e-18], (-N[Sqrt[N[(N[((-F) / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if C < -1.25e-243

   1. Initial program 27.4%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 12.1

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

   8. Applied rewrites 12.1%

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

   if -1.25e-243 < C < 1.40000000000000006e-18

   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 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 43.4%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 21.3

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

   8. Applied rewrites 21.3%

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

   if 1.40000000000000006e-18 < C

   1. Initial program 6.5%

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

   3. Taylor expanded in 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 41.1

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

   7. Applied rewrites 41.1%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{C} \cdot -1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.25 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;-\sqrt{\frac{-F}{B} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \end{array} \end{array} \]

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 (<= C -1e-310) (sqrt (/ (- F) C)) (- (sqrt (* (/ F C) -1.0)))))

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 (C <= -1e-310) {
		tmp = sqrt((-F / C));
	} else {
		tmp = -sqrt(((F / C) * -1.0));
	}
	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 (c <= (-1d-310)) then
        tmp = sqrt((-f / c))
    else
        tmp = -sqrt(((f / c) * (-1.0d0)))
    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 (C <= -1e-310) {
		tmp = Math.sqrt((-F / C));
	} else {
		tmp = -Math.sqrt(((F / C) * -1.0));
	}
	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 C <= -1e-310:
		tmp = math.sqrt((-F / C))
	else:
		tmp = -math.sqrt(((F / C) * -1.0))
	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 (C <= -1e-310)
		tmp = sqrt(Float64(Float64(-F) / C));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / C) * -1.0)));
	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 (C <= -1e-310)
		tmp = sqrt((-F / C));
	else
		tmp = -sqrt(((F / C) * -1.0));
	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[C, -1e-310], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], (-N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if C < -9.999999999999969e-311

   1. Initial program 28.3%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lift-/.f64 12.2

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

   8. Applied rewrites 12.2%

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

   if -9.999999999999969e-311 < C

   1. Initial program 19.7%

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

   3. Taylor expanded in 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 31.1

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

   7. Applied rewrites 31.1%

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

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{C} \cdot -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 2.4% accurate, 18.2× 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 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 24.0%

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

3. Taylor expanded in F around -inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. sqrt-unprod N/A

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

5. Applied rewrites 9.3%

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

6. Taylor expanded in B around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 1.7

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

8. Applied rewrites 1.7%

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

Alternative 9: 20.3% accurate, 20.5× speedup

Math

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

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 (/ (- F) C)))

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((-F / C));
}

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((-f / c))
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((-F / C));
}

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((-F / C))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(Float64(-F) / C))
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((-F / C));
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[((-F) / C), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 24.0%

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

3. Taylor expanded in F around -inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. sqrt-unprod N/A

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

5. Applied rewrites 9.3%

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

6. Taylor expanded in A around -inf

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

   1. lower-*.f64 N/A

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

   2. lift-/.f64 11.4

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

8. Applied rewrites 11.4%

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

9. Final simplification 11.4%

   \[\leadsto \sqrt{\frac{-F}{C}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025071 
(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))))