ABCF->ab-angle b

Percentage Accurate: 18.7% → 46.8%

Time: 12.5s

Alternatives: 11

Speedup: N/A×

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: 46.8% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1400000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - -1 \cdot A\right)}}{-1 \cdot {B\_m}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B\_m}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\_m\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1400000000.0)
   (/
    (sqrt (* (* 2.0 (* (- (pow B_m 2.0) (* (* 4.0 A) C)) F)) (- A (* -1.0 A))))
    (- (* -1.0 (pow B_m 2.0)) (* (* (* -1.0 4.0) A) C)))
   (*
    (* -1.0 (/ (pow 2.0 0.5) B_m))
    (pow (* F (* -1.0 (+ (* -1.0 A) (hypot (* -1.0 A) (* -1.0 B_m))))) 0.5))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1400000000.0) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * (A - (-1.0 * A)))) / ((-1.0 * pow(B_m, 2.0)) - (((-1.0 * 4.0) * A) * C));
	} else {
		tmp = (-1.0 * (pow(2.0, 0.5) / B_m)) * pow((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	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 (B_m <= 1400000000.0) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * (A - (-1.0 * A)))) / ((-1.0 * Math.pow(B_m, 2.0)) - (((-1.0 * 4.0) * A) * C));
	} else {
		tmp = (-1.0 * (Math.pow(2.0, 0.5) / B_m)) * Math.pow((F * (-1.0 * ((-1.0 * A) + Math.hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1400000000.0:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * (A - (-1.0 * A)))) / ((-1.0 * math.pow(B_m, 2.0)) - (((-1.0 * 4.0) * A) * C))
	else:
		tmp = (-1.0 * (math.pow(2.0, 0.5) / B_m)) * math.pow((F * (-1.0 * ((-1.0 * A) + math.hypot((-1.0 * A), (-1.0 * B_m))))), 0.5)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1400000000.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(A - Float64(-1.0 * A)))) / Float64(Float64(-1.0 * (B_m ^ 2.0)) - Float64(Float64(Float64(-1.0 * 4.0) * A) * C)));
	else
		tmp = Float64(Float64(-1.0 * Float64((2.0 ^ 0.5) / B_m)) * (Float64(F * Float64(-1.0 * Float64(Float64(-1.0 * A) + hypot(Float64(-1.0 * A), Float64(-1.0 * B_m))))) ^ 0.5));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1400000000.0)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - ((4.0 * A) * C)) * F)) * (A - (-1.0 * A)))) / ((-1.0 * (B_m ^ 2.0)) - (((-1.0 * 4.0) * A) * C));
	else
		tmp = (-1.0 * ((2.0 ^ 0.5) / B_m)) * ((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1400000000.0], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(-1.0 * N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-1.0 * 4.0), $MachinePrecision] * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Power[2.0, 0.5], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * N[(-1.0 * N[(N[(-1.0 * A), $MachinePrecision] + N[Sqrt[N[(-1.0 * A), $MachinePrecision] ^ 2 + N[(-1.0 * B$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.4e9

   1. Initial program 18.3%

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

   3. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 20.3

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

   5. Applied rewrites 20.3%

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

   if 1.4e9 < B

   1. Initial program 8.7%

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

   3. Taylor expanded in C around 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. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 12.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow1/2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-neg.f64 40.5

         \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]

   7. Applied rewrites 40.5%

      \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1400000000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - -1 \cdot A\right)}}{-1 \cdot {B}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\right)\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.7% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {\left(A - C\right)}^{2}\\ t_1 := -1 \cdot {B\_m}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{t\_0 + {B\_m}^{2}}\right)}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;{\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(\left(A + C\right) - \sqrt{{B\_m}^{2} + t\_0}\right) \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2}}{t\_1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right), 2 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) + A\right)\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B\_m}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\_m\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

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 (- A C) 2.0))
        (t_1 (- (* -1.0 (pow B_m 2.0)) (* (* (* -1.0 4.0) A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) (* (* 4.0 A) C)) F))
            (- (+ A C) (sqrt (+ t_0 (pow B_m 2.0))))))
          t_1)))
   (if (<= t_2 (- INFINITY))
     (* (pow (* -0.5 (/ F C)) 0.5) (* -1.0 (pow 2.0 0.5)))
     (if (<= t_2 -1e-193)
       (/
        (*
         (sqrt
          (*
           F
           (*
            (- (+ A C) (sqrt (+ (pow B_m 2.0) t_0)))
            (- (pow B_m 2.0) (* 4.0 (* A C))))))
         (sqrt 2.0))
        t_1)
       (if (<= t_2 INFINITY)
         (/
          (sqrt
           (fma
            -8.0
            (* A (* C (* F (- A (* -1.0 A)))))
            (* 2.0 (* (* B_m B_m) (* F (+ (+ A (* 2.0 A)) A))))))
          t_1)
         (*
          (* -1.0 (/ (pow 2.0 0.5) B_m))
          (pow
           (* F (* -1.0 (+ (* -1.0 A) (hypot (* -1.0 A) (* -1.0 B_m)))))
           0.5)))))))

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((A - C), 2.0);
	double t_1 = (-1.0 * pow(B_m, 2.0)) - (((-1.0 * 4.0) * A) * C);
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt((t_0 + pow(B_m, 2.0)))))) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = pow((-0.5 * (F / C)), 0.5) * (-1.0 * pow(2.0, 0.5));
	} else if (t_2 <= -1e-193) {
		tmp = (sqrt((F * (((A + C) - sqrt((pow(B_m, 2.0) + t_0))) * (pow(B_m, 2.0) - (4.0 * (A * C)))))) * sqrt(2.0)) / t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(fma(-8.0, (A * (C * (F * (A - (-1.0 * A))))), (2.0 * ((B_m * B_m) * (F * ((A + (2.0 * A)) + A)))))) / t_1;
	} else {
		tmp = (-1.0 * (pow(2.0, 0.5) / B_m)) * pow((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	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(A - C) ^ 2.0
	t_1 = Float64(Float64(-1.0 * (B_m ^ 2.0)) - Float64(Float64(Float64(-1.0 * 4.0) * A) * C))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + C) - sqrt(Float64(t_0 + (B_m ^ 2.0)))))) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64((Float64(-0.5 * Float64(F / C)) ^ 0.5) * Float64(-1.0 * (2.0 ^ 0.5)));
	elseif (t_2 <= -1e-193)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + t_0))) * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C)))))) * sqrt(2.0)) / t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(fma(-8.0, Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A))))), Float64(2.0 * Float64(Float64(B_m * B_m) * Float64(F * Float64(Float64(A + Float64(2.0 * A)) + A)))))) / t_1);
	else
		tmp = Float64(Float64(-1.0 * Float64((2.0 ^ 0.5) / B_m)) * (Float64(F * Float64(-1.0 * Float64(Float64(-1.0 * A) + hypot(Float64(-1.0 * A), Float64(-1.0 * B_m))))) ^ 0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 * N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-1.0 * 4.0), $MachinePrecision] * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(t$95$0 + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[Power[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-193], N[(N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(F * N[(N[(A + N[(2.0 * A), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(-1.0 * N[(N[Power[2.0, 0.5], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * N[(-1.0 * N[(N[(-1.0 * A), $MachinePrecision] + N[Sqrt[N[(-1.0 * A), $MachinePrecision] ^ 2 + N[(-1.0 * B$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.5%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 14.3%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 25.0

         \[\leadsto -1 \cdot \left({\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   8. Applied rewrites 25.0%

      \[\leadsto -1 \cdot \left({\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-193

   1. Initial program 99.2%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   8. Applied rewrites 90.0%

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

   if -1e-193 < (/.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 14.0%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{-\sqrt{\color{blue}{C \cdot \left(-8 \cdot \left(A \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right) + 2 \cdot \frac{F \cdot \left(2 \cdot \left(A \cdot {B}^{2}\right) + {B}^{2} \cdot \left(A - -1 \cdot A\right)\right)}{C}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{C \cdot \color{blue}{\left(-8 \cdot \left(A \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right) + 2 \cdot \frac{F \cdot \left(2 \cdot \left(A \cdot {B}^{2}\right) + {B}^{2} \cdot \left(A - -1 \cdot A\right)\right)}{C}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{C \cdot \mathsf{fma}\left(-8, \color{blue}{A \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)}, 2 \cdot \frac{F \cdot \left(2 \cdot \left(A \cdot {B}^{2}\right) + {B}^{2} \cdot \left(A - -1 \cdot A\right)\right)}{C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{C \cdot \mathsf{fma}\left(-8, A \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}, 2 \cdot \frac{F \cdot \left(2 \cdot \left(A \cdot {B}^{2}\right) + {B}^{2} \cdot \left(A - -1 \cdot A\right)\right)}{C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{C \cdot \mathsf{fma}\left(-8, A \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right), 2 \cdot \frac{F \cdot \left(2 \cdot \left(A \cdot {B}^{2}\right) + {B}^{2} \cdot \left(A - -1 \cdot A\right)\right)}{C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{-\sqrt{C \cdot \mathsf{fma}\left(-8, A \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right), 2 \cdot \frac{F \cdot \left(2 \cdot \left(A \cdot {B}^{2}\right) + {B}^{2} \cdot \left(A - -1 \cdot A\right)\right)}{C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{C \cdot \mathsf{fma}\left(-8, A \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right), 2 \cdot \frac{F \cdot \left(2 \cdot \left(A \cdot {B}^{2}\right) + {B}^{2} \cdot \left(A - -1 \cdot A\right)\right)}{C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{C \cdot \mathsf{fma}\left(-8, A \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right), 2 \cdot \frac{F \cdot \left(2 \cdot \left(A \cdot {B}^{2}\right) + {B}^{2} \cdot \left(A - -1 \cdot A\right)\right)}{C}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 28.5%

      \[\leadsto \frac{-\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(-8, A \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right), 2 \cdot \frac{F \cdot \mathsf{fma}\left(2, A \cdot \left(B \cdot B\right), \left(B \cdot B\right) \cdot \left(A - -1 \cdot A\right)\right)}{C}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right) + \color{blue}{2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}, 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right), 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right), 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right), 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot \color{blue}{A}\right)\right)\right), 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)\right), 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right), 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right), 2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. pow2 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right), 2 \cdot \left(\left(B \cdot B\right) \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right), 2 \cdot \left(\left(B \cdot B\right) \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right), 2 \cdot \left(\left(B \cdot B\right) \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Applied rewrites 33.2%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-8, \color{blue}{A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}, 2 \cdot \left(\left(B \cdot B\right) \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) - -1 \cdot A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in 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. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 1.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow1/2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-neg.f64 14.0

         \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]

   7. Applied rewrites 14.0%

      \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-1 \cdot {B}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;{\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-1 \cdot {B}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C} \leq -1 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2}}{-1 \cdot {B}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-1 \cdot {B}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right), 2 \cdot \left(\left(B \cdot B\right) \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) + A\right)\right)\right)\right)}}{-1 \cdot {B}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\right)\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.7% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \mathsf{fma}\left(-4, A \cdot \left(C \cdot F\right), \left(B\_m \cdot B\_m\right) \cdot F\right)\right) \cdot \left(A - -1 \cdot A\right)}}{-1 \cdot {B\_m}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;{\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B\_m}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\_m\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-117)
   (/
    (sqrt
     (* (* 2.0 (fma -4.0 (* A (* C F)) (* (* B_m B_m) F))) (- A (* -1.0 A))))
    (- (* -1.0 (pow B_m 2.0)) (* (* (* -1.0 4.0) A) C)))
   (if (<= (pow B_m 2.0) 4e-21)
     (* (pow (* -0.5 (/ F C)) 0.5) (* -1.0 (pow 2.0 0.5)))
     (*
      (* -1.0 (/ (pow 2.0 0.5) B_m))
      (pow
       (* F (* -1.0 (+ (* -1.0 A) (hypot (* -1.0 A) (* -1.0 B_m)))))
       0.5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-117) {
		tmp = sqrt(((2.0 * fma(-4.0, (A * (C * F)), ((B_m * B_m) * F))) * (A - (-1.0 * A)))) / ((-1.0 * pow(B_m, 2.0)) - (((-1.0 * 4.0) * A) * C));
	} else if (pow(B_m, 2.0) <= 4e-21) {
		tmp = pow((-0.5 * (F / C)), 0.5) * (-1.0 * pow(2.0, 0.5));
	} else {
		tmp = (-1.0 * (pow(2.0, 0.5) / B_m)) * pow((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-117)
		tmp = Float64(sqrt(Float64(Float64(2.0 * fma(-4.0, Float64(A * Float64(C * F)), Float64(Float64(B_m * B_m) * F))) * Float64(A - Float64(-1.0 * A)))) / Float64(Float64(-1.0 * (B_m ^ 2.0)) - Float64(Float64(Float64(-1.0 * 4.0) * A) * C)));
	elseif ((B_m ^ 2.0) <= 4e-21)
		tmp = Float64((Float64(-0.5 * Float64(F / C)) ^ 0.5) * Float64(-1.0 * (2.0 ^ 0.5)));
	else
		tmp = Float64(Float64(-1.0 * Float64((2.0 ^ 0.5) / B_m)) * (Float64(F * Float64(-1.0 * Float64(Float64(-1.0 * A) + hypot(Float64(-1.0 * A), Float64(-1.0 * B_m))))) ^ 0.5));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-117], N[(N[Sqrt[N[(N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision] + N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(-1.0 * N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-1.0 * 4.0), $MachinePrecision] * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-21], N[(N[Power[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Power[2.0, 0.5], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * N[(-1.0 * N[(N[(-1.0 * A), $MachinePrecision] + N[Sqrt[N[(-1.0 * A), $MachinePrecision] ^ 2 + N[(-1.0 * B$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000003e-117

   1. Initial program 19.9%

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

   3. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 26.7

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

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 22.6

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

   8. Applied rewrites 22.6%

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

   if 1.00000000000000003e-117 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999963e-21

   1. Initial program 22.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lower-*.f64 N/A

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.9

         \[\leadsto -1 \cdot \left({\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   8. Applied rewrites 34.9%

      \[\leadsto -1 \cdot \left({\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   if 3.99999999999999963e-21 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.9%

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

   3. Taylor expanded in 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. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 6.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow1/2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-neg.f64 19.6

         \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]

   7. Applied rewrites 19.6%

      \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \mathsf{fma}\left(-4, A \cdot \left(C \cdot F\right), \left(B \cdot B\right) \cdot F\right)\right) \cdot \left(A - -1 \cdot A\right)}}{-1 \cdot {B}^{2} - \left(\left(-1 \cdot 4\right) \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;{\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\right)\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.3% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;{\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B\_m}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\_m\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e-21)
   (* (pow (* -0.5 (/ F C)) 0.5) (* -1.0 (pow 2.0 0.5)))
   (*
    (* -1.0 (/ (pow 2.0 0.5) B_m))
    (pow (* F (* -1.0 (+ (* -1.0 A) (hypot (* -1.0 A) (* -1.0 B_m))))) 0.5))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 4e-21) {
		tmp = pow((-0.5 * (F / C)), 0.5) * (-1.0 * pow(2.0, 0.5));
	} else {
		tmp = (-1.0 * (pow(2.0, 0.5) / B_m)) * pow((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	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 (Math.pow(B_m, 2.0) <= 4e-21) {
		tmp = Math.pow((-0.5 * (F / C)), 0.5) * (-1.0 * Math.pow(2.0, 0.5));
	} else {
		tmp = (-1.0 * (Math.pow(2.0, 0.5) / B_m)) * Math.pow((F * (-1.0 * ((-1.0 * A) + Math.hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e-21:
		tmp = math.pow((-0.5 * (F / C)), 0.5) * (-1.0 * math.pow(2.0, 0.5))
	else:
		tmp = (-1.0 * (math.pow(2.0, 0.5) / B_m)) * math.pow((F * (-1.0 * ((-1.0 * A) + math.hypot((-1.0 * A), (-1.0 * B_m))))), 0.5)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-21)
		tmp = Float64((Float64(-0.5 * Float64(F / C)) ^ 0.5) * Float64(-1.0 * (2.0 ^ 0.5)));
	else
		tmp = Float64(Float64(-1.0 * Float64((2.0 ^ 0.5) / B_m)) * (Float64(F * Float64(-1.0 * Float64(Float64(-1.0 * A) + hypot(Float64(-1.0 * A), Float64(-1.0 * B_m))))) ^ 0.5));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e-21)
		tmp = ((-0.5 * (F / C)) ^ 0.5) * (-1.0 * (2.0 ^ 0.5));
	else
		tmp = (-1.0 * ((2.0 ^ 0.5) / B_m)) * ((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))) ^ 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-21], N[(N[Power[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Power[2.0, 0.5], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * N[(-1.0 * N[(N[(-1.0 * A), $MachinePrecision] + N[Sqrt[N[(-1.0 * A), $MachinePrecision] ^ 2 + N[(-1.0 * B$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999963e-21

   1. Initial program 20.3%

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

   3. Taylor expanded in F around 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. lower-*.f64 N/A

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 20.2

         \[\leadsto -1 \cdot \left({\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   8. Applied rewrites 20.2%

      \[\leadsto -1 \cdot \left({\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   if 3.99999999999999963e-21 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.9%

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

   3. Taylor expanded in 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. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 6.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow1/2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-neg.f64 19.6

         \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]

   7. Applied rewrites 19.6%

      \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;{\left(-0.5 \cdot \frac{F}{C}\right)}^{0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\right)\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.4% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;{\left(-1 \cdot \frac{F}{B\_m}\right)}^{0.5} \cdot \left(-1 \cdot e^{\log 2 \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B\_m}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\_m\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

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 -5.2e+25)
   (* (pow (* -1.0 (/ F B_m)) 0.5) (* -1.0 (exp (* (log 2.0) 0.5))))
   (*
    (* -1.0 (/ (pow 2.0 0.5) B_m))
    (pow (* F (* -1.0 (+ (* -1.0 A) (hypot (* -1.0 A) (* -1.0 B_m))))) 0.5))))

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 <= -5.2e+25) {
		tmp = pow((-1.0 * (F / B_m)), 0.5) * (-1.0 * exp((log(2.0) * 0.5)));
	} else {
		tmp = (-1.0 * (pow(2.0, 0.5) / B_m)) * pow((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	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 <= -5.2e+25) {
		tmp = Math.pow((-1.0 * (F / B_m)), 0.5) * (-1.0 * Math.exp((Math.log(2.0) * 0.5)));
	} else {
		tmp = (-1.0 * (Math.pow(2.0, 0.5) / B_m)) * Math.pow((F * (-1.0 * ((-1.0 * A) + Math.hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	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 <= -5.2e+25:
		tmp = math.pow((-1.0 * (F / B_m)), 0.5) * (-1.0 * math.exp((math.log(2.0) * 0.5)))
	else:
		tmp = (-1.0 * (math.pow(2.0, 0.5) / B_m)) * math.pow((F * (-1.0 * ((-1.0 * A) + math.hypot((-1.0 * A), (-1.0 * B_m))))), 0.5)
	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 <= -5.2e+25)
		tmp = Float64((Float64(-1.0 * Float64(F / B_m)) ^ 0.5) * Float64(-1.0 * exp(Float64(log(2.0) * 0.5))));
	else
		tmp = Float64(Float64(-1.0 * Float64((2.0 ^ 0.5) / B_m)) * (Float64(F * Float64(-1.0 * Float64(Float64(-1.0 * A) + hypot(Float64(-1.0 * A), Float64(-1.0 * B_m))))) ^ 0.5));
	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 <= -5.2e+25)
		tmp = ((-1.0 * (F / B_m)) ^ 0.5) * (-1.0 * exp((log(2.0) * 0.5)));
	else
		tmp = (-1.0 * ((2.0 ^ 0.5) / B_m)) * ((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))) ^ 0.5);
	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, -5.2e+25], N[(N[Power[N[(-1.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 * N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Power[2.0, 0.5], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * N[(-1.0 * N[(N[(-1.0 * A), $MachinePrecision] + N[Sqrt[N[(-1.0 * A), $MachinePrecision] ^ 2 + N[(-1.0 * B$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.1999999999999997e25

   1. Initial program 11.9%

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

   3. Taylor expanded in F around 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. lower-*.f64 N/A

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 12.8

         \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   8. Applied rewrites 12.8%

      \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 12.8

         \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot e^{\log 2 \cdot 0.5}\right) \]

   10. Applied rewrites 12.8%

       \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot e^{\log 2 \cdot 0.5}\right) \]

   if -5.1999999999999997e25 < F

   1. Initial program 19.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 7.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow1/2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-neg.f64 16.2

         \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]

   7. Applied rewrites 16.2%

      \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;{\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot \left(-1 \cdot e^{\log 2 \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\right)\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.8% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+96}:\\ \;\;\;\;e^{\log \left(-1 \cdot \frac{F}{B\_m}\right) \cdot 0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B\_m}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\_m\right)\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

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 -3.4e+96)
   (* (exp (* (log (* -1.0 (/ F B_m))) 0.5)) (* -1.0 (pow 2.0 0.5)))
   (*
    (* -1.0 (/ (pow 2.0 0.5) B_m))
    (pow (* F (* -1.0 (+ (* -1.0 A) (hypot (* -1.0 A) (* -1.0 B_m))))) 0.5))))

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 <= -3.4e+96) {
		tmp = exp((log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * pow(2.0, 0.5));
	} else {
		tmp = (-1.0 * (pow(2.0, 0.5) / B_m)) * pow((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	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 <= -3.4e+96) {
		tmp = Math.exp((Math.log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * Math.pow(2.0, 0.5));
	} else {
		tmp = (-1.0 * (Math.pow(2.0, 0.5) / B_m)) * Math.pow((F * (-1.0 * ((-1.0 * A) + Math.hypot((-1.0 * A), (-1.0 * B_m))))), 0.5);
	}
	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 <= -3.4e+96:
		tmp = math.exp((math.log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * math.pow(2.0, 0.5))
	else:
		tmp = (-1.0 * (math.pow(2.0, 0.5) / B_m)) * math.pow((F * (-1.0 * ((-1.0 * A) + math.hypot((-1.0 * A), (-1.0 * B_m))))), 0.5)
	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 <= -3.4e+96)
		tmp = Float64(exp(Float64(log(Float64(-1.0 * Float64(F / B_m))) * 0.5)) * Float64(-1.0 * (2.0 ^ 0.5)));
	else
		tmp = Float64(Float64(-1.0 * Float64((2.0 ^ 0.5) / B_m)) * (Float64(F * Float64(-1.0 * Float64(Float64(-1.0 * A) + hypot(Float64(-1.0 * A), Float64(-1.0 * B_m))))) ^ 0.5));
	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 <= -3.4e+96)
		tmp = exp((log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * (2.0 ^ 0.5));
	else
		tmp = (-1.0 * ((2.0 ^ 0.5) / B_m)) * ((F * (-1.0 * ((-1.0 * A) + hypot((-1.0 * A), (-1.0 * B_m))))) ^ 0.5);
	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, -3.4e+96], N[(N[Exp[N[(N[Log[N[(-1.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Power[2.0, 0.5], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * N[(-1.0 * N[(N[(-1.0 * A), $MachinePrecision] + N[Sqrt[N[(-1.0 * A), $MachinePrecision] ^ 2 + N[(-1.0 * B$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.4000000000000001e96

   1. Initial program 13.3%

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

   3. Taylor expanded in F around 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. lower-*.f64 N/A

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

   5. Applied rewrites 14.7%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 14.3

         \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   8. Applied rewrites 14.3%

      \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 13.0

         \[\leadsto -1 \cdot \left(e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot {2}^{0.5}\right) \]

   10. Applied rewrites 13.0%

       \[\leadsto -1 \cdot \left(e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot {\color{blue}{2}}^{0.5}\right) \]

   if -3.4000000000000001e96 < F

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 7.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow1/2 N/A

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

      5. sqr-neg-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-neg.f64 15.7

         \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]

   7. Applied rewrites 15.7%

      \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot {\left(F \cdot \left(A - \mathsf{hypot}\left(-A, -B\right)\right)\right)}^{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+96}:\\ \;\;\;\;e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B}\right) \cdot {\left(F \cdot \left(-1 \cdot \left(-1 \cdot A + \mathsf{hypot}\left(-1 \cdot A, -1 \cdot B\right)\right)\right)\right)}^{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.5% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;e^{\log \left(-1 \cdot \frac{F}{B\_m}\right) \cdot 0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B\_m}\right) \cdot e^{\log \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

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 -5.2e+25)
   (* (exp (* (log (* -1.0 (/ F B_m))) 0.5)) (* -1.0 (pow 2.0 0.5)))
   (*
    (* -1.0 (/ (pow 2.0 0.5) B_m))
    (exp (* (log (* F (- A (hypot A B_m)))) 0.5)))))

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 <= -5.2e+25) {
		tmp = exp((log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * pow(2.0, 0.5));
	} else {
		tmp = (-1.0 * (pow(2.0, 0.5) / B_m)) * exp((log((F * (A - hypot(A, B_m)))) * 0.5));
	}
	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 <= -5.2e+25) {
		tmp = Math.exp((Math.log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * Math.pow(2.0, 0.5));
	} else {
		tmp = (-1.0 * (Math.pow(2.0, 0.5) / B_m)) * Math.exp((Math.log((F * (A - Math.hypot(A, B_m)))) * 0.5));
	}
	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 <= -5.2e+25:
		tmp = math.exp((math.log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * math.pow(2.0, 0.5))
	else:
		tmp = (-1.0 * (math.pow(2.0, 0.5) / B_m)) * math.exp((math.log((F * (A - math.hypot(A, B_m)))) * 0.5))
	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 <= -5.2e+25)
		tmp = Float64(exp(Float64(log(Float64(-1.0 * Float64(F / B_m))) * 0.5)) * Float64(-1.0 * (2.0 ^ 0.5)));
	else
		tmp = Float64(Float64(-1.0 * Float64((2.0 ^ 0.5) / B_m)) * exp(Float64(log(Float64(F * Float64(A - hypot(A, B_m)))) * 0.5)));
	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 <= -5.2e+25)
		tmp = exp((log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * (2.0 ^ 0.5));
	else
		tmp = (-1.0 * ((2.0 ^ 0.5) / B_m)) * exp((log((F * (A - hypot(A, B_m)))) * 0.5));
	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, -5.2e+25], N[(N[Exp[N[(N[Log[N[(-1.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Power[2.0, 0.5], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Log[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.1999999999999997e25

   1. Initial program 11.9%

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

   3. Taylor expanded in F around 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. lower-*.f64 N/A

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

   5. Applied rewrites 14.0%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 12.8

         \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   8. Applied rewrites 12.8%

      \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 11.8

         \[\leadsto -1 \cdot \left(e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot {2}^{0.5}\right) \]

   10. Applied rewrites 11.8%

       \[\leadsto -1 \cdot \left(e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot {\color{blue}{2}}^{0.5}\right) \]

   if -5.1999999999999997e25 < F

   1. Initial program 19.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 7.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow-to-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 15.3%

      \[\leadsto -1 \cdot \left(\frac{{2}^{0.5}}{B} \cdot e^{\log \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right) \cdot 0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+25}:\\ \;\;\;\;e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 \cdot \frac{{2}^{0.5}}{B}\right) \cdot e^{\log \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.6% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{+226}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{-1 \cdot B\_m} \cdot {\left(F \cdot \left(A - {\left(\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)\right)}^{0.5}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(-1 \cdot \frac{F}{B\_m}\right) \cdot 0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \end{array} \end{array} \]

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
 (if (<= (pow B_m 2.0) 1e+226)
   (*
    (/ (exp (* (log 2.0) 0.5)) (* -1.0 B_m))
    (pow (* F (- A (pow (fma A A (* B_m B_m)) 0.5))) 0.5))
   (* (exp (* (log (* -1.0 (/ F B_m))) 0.5)) (* -1.0 (pow 2.0 0.5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e+226) {
		tmp = (exp((log(2.0) * 0.5)) / (-1.0 * B_m)) * pow((F * (A - pow(fma(A, A, (B_m * B_m)), 0.5))), 0.5);
	} else {
		tmp = exp((log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * pow(2.0, 0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+226)
		tmp = Float64(Float64(exp(Float64(log(2.0) * 0.5)) / Float64(-1.0 * B_m)) * (Float64(F * Float64(A - (fma(A, A, Float64(B_m * B_m)) ^ 0.5))) ^ 0.5));
	else
		tmp = Float64(exp(Float64(log(Float64(-1.0 * Float64(F / B_m))) * 0.5)) * Float64(-1.0 * (2.0 ^ 0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+226], N[(N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / N[(-1.0 * B$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * N[(A - N[Power[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[N[(-1.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999961e225

   1. Initial program 22.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 8.1%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 8.1

         \[\leadsto -1 \cdot \left(\frac{e^{\log 2 \cdot 0.5}}{B} \cdot {\left(F \cdot \left(A - {\left(\mathsf{fma}\left(A, A, B \cdot B\right)\right)}^{0.5}\right)\right)}^{0.5}\right) \]

   7. Applied rewrites 8.1%

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

   if 9.99999999999999961e225 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.7%

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

   3. Taylor expanded in 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. lower-*.f64 N/A

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 17.3

         \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   8. Applied rewrites 17.3%

      \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 16.5

         \[\leadsto -1 \cdot \left(e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot {2}^{0.5}\right) \]

   10. Applied rewrites 16.5%

       \[\leadsto -1 \cdot \left(e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot {\color{blue}{2}}^{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+226}:\\ \;\;\;\;\frac{e^{\log 2 \cdot 0.5}}{-1 \cdot B} \cdot {\left(F \cdot \left(A - {\left(\mathsf{fma}\left(A, A, B \cdot B\right)\right)}^{0.5}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.0% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -6 \cdot 10^{+199}:\\ \;\;\;\;-1 \cdot \left(\left(-1 \cdot {\left(A \cdot F\right)}^{0.5}\right) \cdot \frac{-2}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(-1 \cdot \frac{F}{B\_m}\right) \cdot 0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \end{array} \end{array} \]

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
 (if (<= A -6e+199)
   (* -1.0 (* (* -1.0 (pow (* A F) 0.5)) (/ -2.0 B_m)))
   (* (exp (* (log (* -1.0 (/ F B_m))) 0.5)) (* -1.0 (pow 2.0 0.5)))))

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 (A <= -6e+199) {
		tmp = -1.0 * ((-1.0 * pow((A * F), 0.5)) * (-2.0 / B_m));
	} else {
		tmp = exp((log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * pow(2.0, 0.5));
	}
	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 (a <= (-6d+199)) then
        tmp = (-1.0d0) * (((-1.0d0) * ((a * f) ** 0.5d0)) * ((-2.0d0) / b_m))
    else
        tmp = exp((log(((-1.0d0) * (f / b_m))) * 0.5d0)) * ((-1.0d0) * (2.0d0 ** 0.5d0))
    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 (A <= -6e+199) {
		tmp = -1.0 * ((-1.0 * Math.pow((A * F), 0.5)) * (-2.0 / B_m));
	} else {
		tmp = Math.exp((Math.log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * Math.pow(2.0, 0.5));
	}
	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 A <= -6e+199:
		tmp = -1.0 * ((-1.0 * math.pow((A * F), 0.5)) * (-2.0 / B_m))
	else:
		tmp = math.exp((math.log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * math.pow(2.0, 0.5))
	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 (A <= -6e+199)
		tmp = Float64(-1.0 * Float64(Float64(-1.0 * (Float64(A * F) ^ 0.5)) * Float64(-2.0 / B_m)));
	else
		tmp = Float64(exp(Float64(log(Float64(-1.0 * Float64(F / B_m))) * 0.5)) * Float64(-1.0 * (2.0 ^ 0.5)));
	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 (A <= -6e+199)
		tmp = -1.0 * ((-1.0 * ((A * F) ^ 0.5)) * (-2.0 / B_m));
	else
		tmp = exp((log((-1.0 * (F / B_m))) * 0.5)) * (-1.0 * (2.0 ^ 0.5));
	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[A, -6e+199], N[(-1.0 * N[(N[(-1.0 * N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[N[(-1.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -6.0000000000000002e199

   1. Initial program 1.9%

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

   3. Taylor expanded in 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. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 1.7%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. sqrt-unprod N/A

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

      8. metadata-eval N/A

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

      9. sqrt-pow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 7.9

          \[\leadsto -1 \cdot \left(-1 \cdot \left({\left(A \cdot F\right)}^{0.5} \cdot \frac{-2}{B}\right)\right) \]

   8. Applied rewrites 7.9%

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

   if -6.0000000000000002e199 < A

   1. Initial program 18.4%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 17.8%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 10.7

         \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]

   8. Applied rewrites 10.7%

      \[\leadsto -1 \cdot \left({\left(-1 \cdot \frac{F}{B}\right)}^{0.5} \cdot {2}^{0.5}\right) \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 10.0

         \[\leadsto -1 \cdot \left(e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot {2}^{0.5}\right) \]

   10. Applied rewrites 10.0%

       \[\leadsto -1 \cdot \left(e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot {\color{blue}{2}}^{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -6 \cdot 10^{+199}:\\ \;\;\;\;-1 \cdot \left(\left(-1 \cdot {\left(A \cdot F\right)}^{0.5}\right) \cdot \frac{-2}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(-1 \cdot \frac{F}{B}\right) \cdot 0.5} \cdot \left(-1 \cdot {2}^{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 10.5% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)\\ t_1 := F \cdot \left(C - {t\_0}^{0.5}\right)\\ \mathbf{if}\;C \leq 9.5 \cdot 10^{-185}:\\ \;\;\;\;-1 \cdot \left(\left(-1 \cdot {\left(A \cdot F\right)}^{0.5}\right) \cdot \frac{-2}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;A \cdot \mathsf{fma}\left(-1, \frac{{2}^{0.5}}{A \cdot B\_m} \cdot {t\_1}^{0.5}, -0.5 \cdot \left(\left(B\_m \cdot \left({2}^{0.5} \cdot \left(\frac{F \cdot \left(1 - -1 \cdot \left(C \cdot {\left({t\_0}^{-1}\right)}^{0.5}\right)\right)}{B\_m \cdot B\_m} - -4 \cdot \frac{C \cdot t\_1}{{B\_m}^{4}}\right)\right)\right) \cdot {\left({t\_1}^{-1}\right)}^{0.5}\right)\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* C C))) (t_1 (* F (- C (pow t_0 0.5)))))
   (if (<= C 9.5e-185)
     (* -1.0 (* (* -1.0 (pow (* A F) 0.5)) (/ -2.0 B_m)))
     (*
      A
      (fma
       -1.0
       (* (/ (pow 2.0 0.5) (* A B_m)) (pow t_1 0.5))
       (*
        -0.5
        (*
         (*
          B_m
          (*
           (pow 2.0 0.5)
           (-
            (/
             (* F (- 1.0 (* -1.0 (* C (pow (pow t_0 -1.0) 0.5)))))
             (* B_m B_m))
            (* -4.0 (/ (* C t_1) (pow B_m 4.0))))))
         (pow (pow t_1 -1.0) 0.5))))))))

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 = fma(B_m, B_m, (C * C));
	double t_1 = F * (C - pow(t_0, 0.5));
	double tmp;
	if (C <= 9.5e-185) {
		tmp = -1.0 * ((-1.0 * pow((A * F), 0.5)) * (-2.0 / B_m));
	} else {
		tmp = A * fma(-1.0, ((pow(2.0, 0.5) / (A * B_m)) * pow(t_1, 0.5)), (-0.5 * ((B_m * (pow(2.0, 0.5) * (((F * (1.0 - (-1.0 * (C * pow(pow(t_0, -1.0), 0.5))))) / (B_m * B_m)) - (-4.0 * ((C * t_1) / pow(B_m, 4.0)))))) * pow(pow(t_1, -1.0), 0.5))));
	}
	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 = fma(B_m, B_m, Float64(C * C))
	t_1 = Float64(F * Float64(C - (t_0 ^ 0.5)))
	tmp = 0.0
	if (C <= 9.5e-185)
		tmp = Float64(-1.0 * Float64(Float64(-1.0 * (Float64(A * F) ^ 0.5)) * Float64(-2.0 / B_m)));
	else
		tmp = Float64(A * fma(-1.0, Float64(Float64((2.0 ^ 0.5) / Float64(A * B_m)) * (t_1 ^ 0.5)), Float64(-0.5 * Float64(Float64(B_m * Float64((2.0 ^ 0.5) * Float64(Float64(Float64(F * Float64(1.0 - Float64(-1.0 * Float64(C * ((t_0 ^ -1.0) ^ 0.5))))) / Float64(B_m * B_m)) - Float64(-4.0 * Float64(Float64(C * t_1) / (B_m ^ 4.0)))))) * ((t_1 ^ -1.0) ^ 0.5)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * N[(C - N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 9.5e-185], N[(-1.0 * N[(N[(-1.0 * N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(A * N[(-1.0 * N[(N[(N[Power[2.0, 0.5], $MachinePrecision] / N[(A * B$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(B$95$m * N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(N[(N[(F * N[(1.0 - N[(-1.0 * N[(C * N[Power[N[Power[t$95$0, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(N[(C * t$95$1), $MachinePrecision] / N[Power[B$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[t$95$1, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if C < 9.50000000000000042e-185

   1. Initial program 19.5%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. sqrt-unprod N/A

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

      8. metadata-eval N/A

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

      9. sqrt-pow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 4.8

          \[\leadsto -1 \cdot \left(-1 \cdot \left({\left(A \cdot F\right)}^{0.5} \cdot \frac{-2}{B}\right)\right) \]

   8. Applied rewrites 4.8%

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

   if 9.50000000000000042e-185 < C

   1. Initial program 12.0%

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

   3. Taylor expanded in A around 0

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in A around inf

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

   8. Applied rewrites 4.4%

      \[\leadsto A \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{{2}^{0.5}}{A \cdot B} \cdot {\left(F \cdot \left(C - {\left(\mathsf{fma}\left(B, B, C \cdot C\right)\right)}^{0.5}\right)\right)}^{0.5}, -0.5 \cdot \left(\left(B \cdot \left({2}^{0.5} \cdot \left(\frac{F \cdot \left(1 - -1 \cdot \left(C \cdot {\left({\left(\mathsf{fma}\left(B, B, C \cdot C\right)\right)}^{-1}\right)}^{0.5}\right)\right)}{B \cdot B} - -4 \cdot \frac{C \cdot \left(F \cdot \left(C - {\left(\mathsf{fma}\left(B, B, C \cdot C\right)\right)}^{0.5}\right)\right)}{{B}^{4}}\right)\right)\right) \cdot {\left({\left(F \cdot \left(C - {\left(\mathsf{fma}\left(B, B, C \cdot C\right)\right)}^{0.5}\right)\right)}^{-1}\right)}^{0.5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 9.5 \cdot 10^{-185}:\\ \;\;\;\;-1 \cdot \left(\left(-1 \cdot {\left(A \cdot F\right)}^{0.5}\right) \cdot \frac{-2}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;A \cdot \mathsf{fma}\left(-1, \frac{{2}^{0.5}}{A \cdot B} \cdot {\left(F \cdot \left(C - {\left(\mathsf{fma}\left(B, B, C \cdot C\right)\right)}^{0.5}\right)\right)}^{0.5}, -0.5 \cdot \left(\left(B \cdot \left({2}^{0.5} \cdot \left(\frac{F \cdot \left(1 - -1 \cdot \left(C \cdot {\left({\left(\mathsf{fma}\left(B, B, C \cdot C\right)\right)}^{-1}\right)}^{0.5}\right)\right)}{B \cdot B} - -4 \cdot \frac{C \cdot \left(F \cdot \left(C - {\left(\mathsf{fma}\left(B, B, C \cdot C\right)\right)}^{0.5}\right)\right)}{{B}^{4}}\right)\right)\right) \cdot {\left({\left(F \cdot \left(C - {\left(\mathsf{fma}\left(B, B, C \cdot C\right)\right)}^{0.5}\right)\right)}^{-1}\right)}^{0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 9.8% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)\\ t_1 := F \cdot \left(C - {t\_0}^{0.5}\right)\\ A \cdot \mathsf{fma}\left(-1, \frac{{2}^{0.5}}{A \cdot B\_m} \cdot {t\_1}^{0.5}, -0.5 \cdot \left(\left(B\_m \cdot \left({2}^{0.5} \cdot \left(\frac{F \cdot \left(1 - -1 \cdot \left(C \cdot {\left({t\_0}^{-1}\right)}^{0.5}\right)\right)}{B\_m \cdot B\_m} - -4 \cdot \frac{C \cdot t\_1}{{B\_m}^{4}}\right)\right)\right) \cdot {\left({t\_1}^{-1}\right)}^{0.5}\right)\right) \end{array} \end{array} \]

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 (fma B_m B_m (* C C))) (t_1 (* F (- C (pow t_0 0.5)))))
   (*
    A
    (fma
     -1.0
     (* (/ (pow 2.0 0.5) (* A B_m)) (pow t_1 0.5))
     (*
      -0.5
      (*
       (*
        B_m
        (*
         (pow 2.0 0.5)
         (-
          (/ (* F (- 1.0 (* -1.0 (* C (pow (pow t_0 -1.0) 0.5))))) (* B_m B_m))
          (* -4.0 (/ (* C t_1) (pow B_m 4.0))))))
       (pow (pow t_1 -1.0) 0.5)))))))

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 = fma(B_m, B_m, (C * C));
	double t_1 = F * (C - pow(t_0, 0.5));
	return A * fma(-1.0, ((pow(2.0, 0.5) / (A * B_m)) * pow(t_1, 0.5)), (-0.5 * ((B_m * (pow(2.0, 0.5) * (((F * (1.0 - (-1.0 * (C * pow(pow(t_0, -1.0), 0.5))))) / (B_m * B_m)) - (-4.0 * ((C * t_1) / pow(B_m, 4.0)))))) * pow(pow(t_1, -1.0), 0.5))));
}

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 = fma(B_m, B_m, Float64(C * C))
	t_1 = Float64(F * Float64(C - (t_0 ^ 0.5)))
	return Float64(A * fma(-1.0, Float64(Float64((2.0 ^ 0.5) / Float64(A * B_m)) * (t_1 ^ 0.5)), Float64(-0.5 * Float64(Float64(B_m * Float64((2.0 ^ 0.5) * Float64(Float64(Float64(F * Float64(1.0 - Float64(-1.0 * Float64(C * ((t_0 ^ -1.0) ^ 0.5))))) / Float64(B_m * B_m)) - Float64(-4.0 * Float64(Float64(C * t_1) / (B_m ^ 4.0)))))) * ((t_1 ^ -1.0) ^ 0.5)))))
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[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * N[(C - N[Power[t$95$0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(A * N[(-1.0 * N[(N[(N[Power[2.0, 0.5], $MachinePrecision] / N[(A * B$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, 0.5], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(B$95$m * N[(N[Power[2.0, 0.5], $MachinePrecision] * N[(N[(N[(F * N[(1.0 - N[(-1.0 * N[(C * N[Power[N[Power[t$95$0, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(N[(C * t$95$1), $MachinePrecision] / N[Power[B$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[t$95$1, -1.0], $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in A around 0

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

5. Applied rewrites 3.2%

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

6. Taylor expanded in A around inf

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

8. Applied rewrites 3.2%

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

Reproduce

herbie shell --seed 2025065 
(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))))