ABCF->ab-angle a

Percentage Accurate: 19.5% → 55.1%

Time: 14.9s

Alternatives: 13

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: 19.5% 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: 55.1% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := {B\_m}^{2} - t\_0\\ \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)} \cdot \left(-\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (- (pow B_m 2.0) t_0)))
   (if (<= B_m 6.5e-77)
     (/
      (sqrt (* (* 2.0 (* t_1 F)) (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))))
      (- t_1))
     (if (<= B_m 2.6e+69)
       (/
        (*
         (sqrt (* 2.0 (* (- (* B_m B_m) t_0) F)))
         (- (sqrt (+ (+ A C) (hypot (- A C) B_m)))))
        t_1)
       (* (/ (sqrt F) (sqrt B_m)) (- (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = pow(B_m, 2.0) - t_0;
	double tmp;
	if (B_m <= 6.5e-77) {
		tmp = sqrt(((2.0 * (t_1 * F)) * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / -t_1;
	} else if (B_m <= 2.6e+69) {
		tmp = (sqrt((2.0 * (((B_m * B_m) - t_0) * F))) * -sqrt(((A + C) + hypot((A - C), B_m)))) / t_1;
	} else {
		tmp = (sqrt(F) / sqrt(B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64((B_m ^ 2.0) - t_0)
	tmp = 0.0
	if (B_m <= 6.5e-77)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))) / Float64(-t_1));
	elseif (B_m <= 2.6e+69)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F))) * Float64(-sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(F) / sqrt(B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[B$95$m, 6.5e-77], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[B$95$m, 2.6e+69], N[(N[(N[Sqrt[N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.4999999999999999e-77

   1. Initial program 20.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 18.9

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

   5. Applied rewrites 18.9%

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

   if 6.4999999999999999e-77 < B < 2.6000000000000002e69

   1. Initial program 42.8%

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

   4. Applied rewrites 45.8%

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

   if 2.6000000000000002e69 < B

   1. Initial program 9.2%

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

   3. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 59.5

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

   5. Applied rewrites 59.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 59.1

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

   7. Applied rewrites 59.1%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 70.4

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

   9. Applied rewrites 70.4%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \left(-\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.5% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+138}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= (pow B_m 2.0) 5e-132)
     (/
      (sqrt (* (* 2.0 (* t_0 F)) (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))))
      (- t_0))
     (if (<= (pow B_m 2.0) 2e+138)
       (-
        (sqrt
         (*
          (/
           (* F (+ A (+ C (hypot B_m (- A C)))))
           (- (* B_m B_m) (* 4.0 (* A C))))
          2.0)))
       (* (/ (sqrt F) (sqrt B_m)) (- (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-132) {
		tmp = sqrt(((2.0 * (t_0 * F)) * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 2e+138) {
		tmp = -sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else {
		tmp = (sqrt(F) / sqrt(B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-132)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 2e+138)
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C)))) * 2.0)));
	else
		tmp = Float64(Float64(sqrt(F) / sqrt(B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-132], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+138], (-N[Sqrt[N[(N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-132

   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 A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 32.3

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

   5. Applied rewrites 32.3%

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

   if 4.9999999999999999e-132 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e138

   1. Initial program 38.6%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 49.3%

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

   if 2.0000000000000001e138 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.5%

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

   3. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.9

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

   5. Applied rewrites 31.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 31.7

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

   7. Applied rewrites 31.7%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 37.1

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

   9. Applied rewrites 37.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+138}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.4% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(C \cdot C\right) \cdot F\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := -t\_1\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-198}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B\_m \cdot B\_m\right) \cdot \mathsf{fma}\left(-2, t\_0, 0.5 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot t\_0\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(A, B\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* C C) F))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2 (- t_1))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_2)))
   (if (<= t_3 -1e-198)
     (-
      (sqrt
       (*
        (/
         (* F (+ A (+ C (hypot B_m (- A C)))))
         (- (* B_m B_m) (* 4.0 (* A C))))
        2.0)))
     (if (<= t_3 0.0)
       (/
        (sqrt
         (*
          (- A)
          (fma
           -1.0
           (/
            (fma
             -2.0
             (/ (* (* B_m B_m) (fma -2.0 t_0 (* 0.5 (* (* B_m B_m) F)))) A)
             (* 8.0 (* (* B_m B_m) (* C F))))
            A)
           (* 16.0 t_0))))
        t_2)
       (if (<= t_3 INFINITY)
         (sqrt (/ (- F) A))
         (*
          (/ (sqrt 2.0) (- B_m))
          (* (sqrt F) (sqrt (+ A (hypot A B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (C * C) * F;
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -t_1;
	double t_3 = sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -1e-198) {
		tmp = -sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_3 <= 0.0) {
		tmp = sqrt((-A * fma(-1.0, (fma(-2.0, (((B_m * B_m) * fma(-2.0, t_0, (0.5 * ((B_m * B_m) * F)))) / A), (8.0 * ((B_m * B_m) * (C * F)))) / A), (16.0 * t_0)))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((-F / A));
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((A + hypot(A, B_m))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(C * C) * F)
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(-t_1)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= -1e-198)
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C)))) * 2.0)));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(-A) * fma(-1.0, Float64(fma(-2.0, Float64(Float64(Float64(B_m * B_m) * fma(-2.0, t_0, Float64(0.5 * Float64(Float64(B_m * B_m) * F)))) / A), Float64(8.0 * Float64(Float64(B_m * B_m) * Float64(C * F)))) / A), Float64(16.0 * t_0)))) / t_2);
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(-F) / A));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(A + hypot(A, B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-198], (-N[Sqrt[N[(N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[((-A) * N[(-1.0 * N[(N[(-2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(-2.0 * t$95$0 + N[(0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision] + N[(8.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision] + N[(16.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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))) < -9.9999999999999991e-199

   1. Initial program 40.7%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 63.5%

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

   if -9.9999999999999991e-199 < (/.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))) < -0.0

   1. Initial program 7.9%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\color{blue}{-1 \cdot \left(A \cdot \left(-1 \cdot \frac{-2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot {C}^{2}\right) + \frac{1}{2} \cdot {B}^{4}\right)}{A} + 8 \cdot \left({B}^{2} \cdot \left(C \cdot F\right)\right)}{A} + 16 \cdot \left({C}^{2} \cdot F\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 17.8%

      \[\leadsto \frac{-\sqrt{\color{blue}{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{F \cdot \mathsf{fma}\left(-2, {\left(B \cdot C\right)}^{2}, 0.5 \cdot {B}^{4}\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{{B}^{2} \cdot \left(-2 \cdot \left({C}^{2} \cdot F\right) + \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{{B}^{2} \cdot \left(-2 \cdot \left({C}^{2} \cdot F\right) + \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. pow2 N/A

         \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \left(-2 \cdot \left({C}^{2} \cdot F\right) + \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \left(-2 \cdot \left({C}^{2} \cdot F\right) + \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, {C}^{2} \cdot F, \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. pow2 N/A

         \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, \left(C \cdot C\right) \cdot F, \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, \left(C \cdot C\right) \cdot F, \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, \left(C \cdot C\right) \cdot F, \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, \left(C \cdot C\right) \cdot F, \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, \left(C \cdot C\right) \cdot F, \frac{1}{2} \cdot \left({B}^{2} \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. pow2 N/A

          \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, \left(C \cdot C\right) \cdot F, \frac{1}{2} \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. lift-*.f64 21.7

          \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, \left(C \cdot C\right) \cdot F, 0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Applied rewrites 21.7%

      \[\leadsto \frac{-\sqrt{-1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, \left(C \cdot C\right) \cdot F, 0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -0.0 < (/.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))) < +inf.0

   1. Initial program 54.3%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 68.1

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

   8. Applied rewrites 68.1%

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqr-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. 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. lift-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. pow2 N/A

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

      10. lower-hypot.f64 18.9

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

   7. Applied rewrites 18.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 31.4

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

   9. Applied rewrites 31.4%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-198}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-2, \frac{\left(B \cdot B\right) \cdot \mathsf{fma}\left(-2, \left(C \cdot C\right) \cdot F, 0.5 \cdot \left(\left(B \cdot B\right) \cdot F\right)\right)}{A}, 8 \cdot \left(\left(B \cdot B\right) \cdot \left(C \cdot F\right)\right)\right)}{A}, 16 \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(A, B\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.4% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-216}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(A, B\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0))))
   (if (<= t_1 -4e-216)
     (-
      (sqrt
       (*
        (/
         (* F (+ A (+ C (hypot B_m (- A C)))))
         (- (* B_m B_m) (* 4.0 (* A C))))
        2.0)))
     (if (<= t_1 INFINITY)
       (sqrt (/ (- F) A))
       (* (/ (sqrt 2.0) (- B_m)) (* (sqrt F) (sqrt (+ A (hypot A B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if (t_1 <= -4e-216) {
		tmp = -sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((-F / A));
	} else {
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((A + hypot(A, B_m))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if (t_1 <= -4e-216) {
		tmp = -Math.sqrt((((F * (A + (C + Math.hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((-F / A));
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * (Math.sqrt(F) * Math.sqrt((A + Math.hypot(A, B_m))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_1 = math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_0
	tmp = 0
	if t_1 <= -4e-216:
		tmp = -math.sqrt((((F * (A + (C + math.hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0))
	elif t_1 <= math.inf:
		tmp = math.sqrt((-F / A))
	else:
		tmp = (math.sqrt(2.0) / -B_m) * (math.sqrt(F) * math.sqrt((A + math.hypot(A, B_m))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	tmp = 0.0
	if (t_1 <= -4e-216)
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C)))) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(-F) / A));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * Float64(sqrt(F) * sqrt(Float64(A + hypot(A, B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_0;
	tmp = 0.0;
	if (t_1 <= -4e-216)
		tmp = -sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	elseif (t_1 <= Inf)
		tmp = sqrt((-F / A));
	else
		tmp = (sqrt(2.0) / -B_m) * (sqrt(F) * sqrt((A + hypot(A, B_m))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-216], (-N[Sqrt[N[(N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$1, Infinity], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000002e-216

   1. Initial program 41.1%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 62.4%

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

   if -4.0000000000000002e-216 < (/.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 23.0%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 37.8

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

   8. Applied rewrites 37.8%

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

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqr-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-fma.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. 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. lift-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. pow2 N/A

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

      10. lower-hypot.f64 18.9

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

   7. Applied rewrites 18.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 31.4

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

   9. Applied rewrites 31.4%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-216}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(A, B\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.1% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-216}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0))))
   (if (<= t_1 -4e-216)
     (-
      (sqrt
       (*
        (/
         (* F (+ A (+ C (hypot B_m (- A C)))))
         (- (* B_m B_m) (* 4.0 (* A C))))
        2.0)))
     (if (<= t_1 INFINITY)
       (sqrt (/ (- F) A))
       (* (/ (sqrt F) (sqrt B_m)) (- (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if (t_1 <= -4e-216) {
		tmp = -sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((-F / A));
	} else {
		tmp = (sqrt(F) / sqrt(B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if (t_1 <= -4e-216) {
		tmp = -Math.sqrt((((F * (A + (C + Math.hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((-F / A));
	} else {
		tmp = (Math.sqrt(F) / Math.sqrt(B_m)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_1 = math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_0
	tmp = 0
	if t_1 <= -4e-216:
		tmp = -math.sqrt((((F * (A + (C + math.hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0))
	elif t_1 <= math.inf:
		tmp = math.sqrt((-F / A))
	else:
		tmp = (math.sqrt(F) / math.sqrt(B_m)) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	tmp = 0.0
	if (t_1 <= -4e-216)
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C)))) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(-F) / A));
	else
		tmp = Float64(Float64(sqrt(F) / sqrt(B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_0;
	tmp = 0.0;
	if (t_1 <= -4e-216)
		tmp = -sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	elseif (t_1 <= Inf)
		tmp = sqrt((-F / A));
	else
		tmp = (sqrt(F) / sqrt(B_m)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-216], (-N[Sqrt[N[(N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$1, Infinity], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000002e-216

   1. Initial program 41.1%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 62.4%

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

   if -4.0000000000000002e-216 < (/.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 23.0%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 37.8

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

   8. Applied rewrites 37.8%

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

   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 B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 23.8

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

   5. Applied rewrites 23.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 23.7

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

   7. Applied rewrites 23.7%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 28.4

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

   9. Applied rewrites 28.4%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-216}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.2% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -1e-310)
   (sqrt (/ (- F) A))
   (if (<= F 8.2e+68)
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (+ C (hypot B_m C)))))
     (- (sqrt (* (/ F B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1e-310) {
		tmp = sqrt((-F / A));
	} else if (F <= 8.2e+68) {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1e-310) {
		tmp = Math.sqrt((-F / A));
	} else if (F <= 8.2e+68) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (C + Math.hypot(B_m, C))));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -1e-310:
		tmp = math.sqrt((-F / A))
	elif F <= 8.2e+68:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (C + math.hypot(B_m, C))))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -1e-310)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (F <= 8.2e+68)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(C + hypot(B_m, C)))));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -1e-310)
		tmp = sqrt((-F / A));
	elseif (F <= 8.2e+68)
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (C + hypot(B_m, C))));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -1e-310], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 8.2e+68], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if F < -9.999999999999969e-311

   1. Initial program 34.0%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 60.9

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

   8. Applied rewrites 60.9%

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

   if -9.999999999999969e-311 < F < 8.1999999999999998e68

   1. Initial program 25.4%

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

   3. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 26.8

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

   5. Applied rewrites 26.8%

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

   if 8.1999999999999998e68 < F

   1. Initial program 11.8%

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

   3. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 24.0

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

   5. Applied rewrites 24.0%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.6% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-216}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0))))
   (if (<= t_1 -4e-216)
     (-
      (sqrt
       (*
        (/
         (* F (+ A (+ C (hypot B_m (- A C)))))
         (- (* B_m B_m) (* 4.0 (* A C))))
        2.0)))
     (if (<= t_1 INFINITY) (sqrt (/ (- F) A)) (- (sqrt (* (/ F B_m) 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if (t_1 <= -4e-216) {
		tmp = -sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((-F / A));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if (t_1 <= -4e-216) {
		tmp = -Math.sqrt((((F * (A + (C + Math.hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((-F / A));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_1 = math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_0
	tmp = 0
	if t_1 <= -4e-216:
		tmp = -math.sqrt((((F * (A + (C + math.hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0))
	elif t_1 <= math.inf:
		tmp = math.sqrt((-F / A))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	tmp = 0.0
	if (t_1 <= -4e-216)
		tmp = Float64(-sqrt(Float64(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C)))) * 2.0)));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(-F) / A));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_0;
	tmp = 0.0;
	if (t_1 <= -4e-216)
		tmp = -sqrt((((F * (A + (C + hypot(B_m, (A - C))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	elseif (t_1 <= Inf)
		tmp = sqrt((-F / A));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-216], (-N[Sqrt[N[(N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$1, Infinity], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000002e-216

   1. Initial program 41.1%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 62.4%

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

   if -4.0000000000000002e-216 < (/.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 23.0%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 37.8

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

   8. Applied rewrites 37.8%

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

   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 B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 23.8

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

   5. Applied rewrites 23.8%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-216}:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.9% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-57}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B\_m \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\frac{1}{\sqrt{{B\_m}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -1e-310)
   (sqrt (/ (- F) A))
   (if (<= F 8.2e-57)
     (*
      F
      (fma
       -1.0
       (sqrt (* (pow (* B_m F) -1.0) 2.0))
       (* -0.5 (* (/ 1.0 (sqrt (* (pow B_m 3.0) F))) (* (sqrt 2.0) (+ A C))))))
     (- (sqrt (* (/ F B_m) 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1e-310) {
		tmp = sqrt((-F / A));
	} else if (F <= 8.2e-57) {
		tmp = F * fma(-1.0, sqrt((pow((B_m * F), -1.0) * 2.0)), (-0.5 * ((1.0 / sqrt((pow(B_m, 3.0) * F))) * (sqrt(2.0) * (A + C)))));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -1e-310)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (F <= 8.2e-57)
		tmp = Float64(F * fma(-1.0, sqrt(Float64((Float64(B_m * F) ^ -1.0) * 2.0)), Float64(-0.5 * Float64(Float64(1.0 / sqrt(Float64((B_m ^ 3.0) * F))) * Float64(sqrt(2.0) * Float64(A + C))))));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -1e-310], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 8.2e-57], N[(F * N[(-1.0 * N[Sqrt[N[(N[Power[N[(B$95$m * F), $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[(N[(1.0 / N[Sqrt[N[(N[Power[B$95$m, 3.0], $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if F < -9.999999999999969e-311

   1. Initial program 34.0%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 60.9

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

   8. Applied rewrites 60.9%

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

   if -9.999999999999969e-311 < F < 8.2000000000000003e-57

   1. Initial program 28.1%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      13. lift-+.f64 11.3

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   5. Applied rewrites 11.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   6. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{B \cdot F}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{B \cdot F}} \cdot \sqrt{2}\right) + \color{blue}{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F}} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. sqrt-unprod N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   8. Applied rewrites 18.5%

      \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\frac{1}{\sqrt{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   if 8.2000000000000003e-57 < F

   1. Initial program 14.5%

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

   3. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 25.7

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

   5. Applied rewrites 25.7%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-57}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\frac{1}{\sqrt{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.3% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{F}{{B\_m}^{3}}\\ \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+68}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B\_m \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\frac{1}{\sqrt{{B\_m}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B\_m} \cdot 2}, -0.5 \cdot \left(\sqrt{t\_0} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{t\_0 \cdot 2}\right)\\ \end{array} \end{array} \]

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 (/ F (pow B_m 3.0))))
   (if (<= F -1e-310)
     (sqrt (/ (- F) A))
     (if (<= F 8e+68)
       (*
        F
        (fma
         -1.0
         (sqrt (* (pow (* B_m F) -1.0) 2.0))
         (*
          -0.5
          (* (/ 1.0 (sqrt (* (pow B_m 3.0) F))) (* (sqrt 2.0) (+ A C))))))
       (*
        (- A)
        (fma
         -1.0
         (/
          (fma
           -1.0
           (sqrt (* (/ F B_m) 2.0))
           (* -0.5 (* (sqrt t_0) (* C (sqrt 2.0)))))
          A)
         (* 0.5 (sqrt (* t_0 2.0)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = F / pow(B_m, 3.0);
	double tmp;
	if (F <= -1e-310) {
		tmp = sqrt((-F / A));
	} else if (F <= 8e+68) {
		tmp = F * fma(-1.0, sqrt((pow((B_m * F), -1.0) * 2.0)), (-0.5 * ((1.0 / sqrt((pow(B_m, 3.0) * F))) * (sqrt(2.0) * (A + C)))));
	} else {
		tmp = -A * fma(-1.0, (fma(-1.0, sqrt(((F / B_m) * 2.0)), (-0.5 * (sqrt(t_0) * (C * sqrt(2.0))))) / A), (0.5 * sqrt((t_0 * 2.0))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(F / (B_m ^ 3.0))
	tmp = 0.0
	if (F <= -1e-310)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (F <= 8e+68)
		tmp = Float64(F * fma(-1.0, sqrt(Float64((Float64(B_m * F) ^ -1.0) * 2.0)), Float64(-0.5 * Float64(Float64(1.0 / sqrt(Float64((B_m ^ 3.0) * F))) * Float64(sqrt(2.0) * Float64(A + C))))));
	else
		tmp = Float64(Float64(-A) * fma(-1.0, Float64(fma(-1.0, sqrt(Float64(Float64(F / B_m) * 2.0)), Float64(-0.5 * Float64(sqrt(t_0) * Float64(C * sqrt(2.0))))) / A), Float64(0.5 * sqrt(Float64(t_0 * 2.0)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(F / N[Power[B$95$m, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-310], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 8e+68], N[(F * N[(-1.0 * N[Sqrt[N[(N[Power[N[(B$95$m * F), $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[(N[(1.0 / N[Sqrt[N[(N[Power[B$95$m, 3.0], $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-A) * N[(-1.0 * N[(N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(C * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.999999999999969e-311

   1. Initial program 34.0%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 60.9

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

   8. Applied rewrites 60.9%

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

   if -9.999999999999969e-311 < F < 7.99999999999999962e68

   1. Initial program 25.4%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      13. lift-+.f64 17.1

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   5. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   6. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{B \cdot F}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{B \cdot F}} \cdot \sqrt{2}\right) + \color{blue}{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F}} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. sqrt-unprod N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   8. Applied rewrites 21.4%

      \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\frac{1}{\sqrt{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   if 7.99999999999999962e68 < F

   1. Initial program 11.8%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      13. lift-+.f64 23.0

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   5. Applied rewrites 23.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   6. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{\color{blue}{A}}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right) \]

   8. Applied rewrites 20.9%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+68}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\frac{1}{\sqrt{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{\frac{F}{{B}^{3}} \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.4% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{F}{{B\_m}^{3}}\\ \mathbf{if}\;F \leq 8 \cdot 10^{+68}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B\_m \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\sqrt{{\left({B\_m}^{3} \cdot F\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B\_m} \cdot 2}, -0.5 \cdot \left(\sqrt{t\_0} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{t\_0 \cdot 2}\right)\\ \end{array} \end{array} \]

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 (/ F (pow B_m 3.0))))
   (if (<= F 8e+68)
     (*
      F
      (fma
       -1.0
       (sqrt (* (pow (* B_m F) -1.0) 2.0))
       (*
        -0.5
        (* (sqrt (pow (* (pow B_m 3.0) F) -1.0)) (* (sqrt 2.0) (+ A C))))))
     (*
      (- A)
      (fma
       -1.0
       (/
        (fma
         -1.0
         (sqrt (* (/ F B_m) 2.0))
         (* -0.5 (* (sqrt t_0) (* C (sqrt 2.0)))))
        A)
       (* 0.5 (sqrt (* t_0 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = F / pow(B_m, 3.0);
	double tmp;
	if (F <= 8e+68) {
		tmp = F * fma(-1.0, sqrt((pow((B_m * F), -1.0) * 2.0)), (-0.5 * (sqrt(pow((pow(B_m, 3.0) * F), -1.0)) * (sqrt(2.0) * (A + C)))));
	} else {
		tmp = -A * fma(-1.0, (fma(-1.0, sqrt(((F / B_m) * 2.0)), (-0.5 * (sqrt(t_0) * (C * sqrt(2.0))))) / A), (0.5 * sqrt((t_0 * 2.0))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(F / (B_m ^ 3.0))
	tmp = 0.0
	if (F <= 8e+68)
		tmp = Float64(F * fma(-1.0, sqrt(Float64((Float64(B_m * F) ^ -1.0) * 2.0)), Float64(-0.5 * Float64(sqrt((Float64((B_m ^ 3.0) * F) ^ -1.0)) * Float64(sqrt(2.0) * Float64(A + C))))));
	else
		tmp = Float64(Float64(-A) * fma(-1.0, Float64(fma(-1.0, sqrt(Float64(Float64(F / B_m) * 2.0)), Float64(-0.5 * Float64(sqrt(t_0) * Float64(C * sqrt(2.0))))) / A), Float64(0.5 * sqrt(Float64(t_0 * 2.0)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(F / N[Power[B$95$m, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 8e+68], N[(F * N[(-1.0 * N[Sqrt[N[(N[Power[N[(B$95$m * F), $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[(N[Sqrt[N[Power[N[(N[Power[B$95$m, 3.0], $MachinePrecision] * F), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-A) * N[(-1.0 * N[(N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(C * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < 7.99999999999999962e68

   1. Initial program 26.8%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      13. lift-+.f64 14.4

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   5. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   6. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{\color{blue}{A}}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right) \]

   8. Applied rewrites 13.6%

      \[\leadsto -1 \cdot \color{blue}{\left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{\frac{F}{{B}^{3}} \cdot 2}\right)\right)} \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{B \cdot F}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. flip3-+ N/A

          \[\leadsto F \cdot \left(\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{B \cdot F}} \cdot \sqrt{2}\right)} + \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{B \cdot F}} \cdot \sqrt{2}\right) + \color{blue}{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F}} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   11. Applied rewrites 18.4%

       \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\sqrt{{\left({B}^{3} \cdot F\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   if 7.99999999999999962e68 < F

   1. Initial program 11.8%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      13. lift-+.f64 23.0

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   5. Applied rewrites 23.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   6. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{\color{blue}{A}}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right) \]

   8. Applied rewrites 20.9%

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

4. Final simplification 19.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 8 \cdot 10^{+68}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\sqrt{{\left({B}^{3} \cdot F\right)}^{-1}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{\frac{F}{{B}^{3}} \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.5% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{F}{{B\_m}^{3}}\\ \mathbf{if}\;F \leq 8 \cdot 10^{+68}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B\_m \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\frac{1}{\sqrt{{B\_m}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B\_m} \cdot 2}, -0.5 \cdot \left(\sqrt{t\_0} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{t\_0 \cdot 2}\right)\\ \end{array} \end{array} \]

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 (/ F (pow B_m 3.0))))
   (if (<= F 8e+68)
     (*
      F
      (fma
       -1.0
       (sqrt (* (pow (* B_m F) -1.0) 2.0))
       (* -0.5 (* (/ 1.0 (sqrt (* (pow B_m 3.0) F))) (* (sqrt 2.0) (+ A C))))))
     (*
      (- A)
      (fma
       -1.0
       (/
        (fma
         -1.0
         (sqrt (* (/ F B_m) 2.0))
         (* -0.5 (* (sqrt t_0) (* C (sqrt 2.0)))))
        A)
       (* 0.5 (sqrt (* t_0 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = F / pow(B_m, 3.0);
	double tmp;
	if (F <= 8e+68) {
		tmp = F * fma(-1.0, sqrt((pow((B_m * F), -1.0) * 2.0)), (-0.5 * ((1.0 / sqrt((pow(B_m, 3.0) * F))) * (sqrt(2.0) * (A + C)))));
	} else {
		tmp = -A * fma(-1.0, (fma(-1.0, sqrt(((F / B_m) * 2.0)), (-0.5 * (sqrt(t_0) * (C * sqrt(2.0))))) / A), (0.5 * sqrt((t_0 * 2.0))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(F / (B_m ^ 3.0))
	tmp = 0.0
	if (F <= 8e+68)
		tmp = Float64(F * fma(-1.0, sqrt(Float64((Float64(B_m * F) ^ -1.0) * 2.0)), Float64(-0.5 * Float64(Float64(1.0 / sqrt(Float64((B_m ^ 3.0) * F))) * Float64(sqrt(2.0) * Float64(A + C))))));
	else
		tmp = Float64(Float64(-A) * fma(-1.0, Float64(fma(-1.0, sqrt(Float64(Float64(F / B_m) * 2.0)), Float64(-0.5 * Float64(sqrt(t_0) * Float64(C * sqrt(2.0))))) / A), Float64(0.5 * sqrt(Float64(t_0 * 2.0)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(F / N[Power[B$95$m, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 8e+68], N[(F * N[(-1.0 * N[Sqrt[N[(N[Power[N[(B$95$m * F), $MachinePrecision], -1.0], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[(N[(1.0 / N[Sqrt[N[(N[Power[B$95$m, 3.0], $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-A) * N[(-1.0 * N[(N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(C * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < 7.99999999999999962e68

   1. Initial program 26.8%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      13. lift-+.f64 14.4

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   5. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   6. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{\frac{1}{B \cdot F}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{B \cdot F}} \cdot \sqrt{2}\right) + \color{blue}{\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F}} \cdot \color{blue}{\sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. sqrt-unprod N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{\frac{1}{B \cdot F} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. inv-pow N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{1}{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   8. Applied rewrites 18.3%

      \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\frac{1}{\sqrt{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   if 7.99999999999999962e68 < F

   1. Initial program 11.8%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      10. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

      13. lift-+.f64 23.0

          \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   5. Applied rewrites 23.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

   6. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{\color{blue}{A}}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right) \]

   8. Applied rewrites 20.9%

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

4. Final simplification 19.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 8 \cdot 10^{+68}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \sqrt{{\left(B \cdot F\right)}^{-1} \cdot 2}, -0.5 \cdot \left(\frac{1}{\sqrt{{B}^{3} \cdot F}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{\frac{F}{{B}^{3}} \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 23.3% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{F}{{B\_m}^{3}}\\ \left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B\_m} \cdot 2}, -0.5 \cdot \left(\sqrt{t\_0} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{t\_0 \cdot 2}\right) \end{array} \end{array} \]

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 (/ F (pow B_m 3.0))))
   (*
    (- A)
    (fma
     -1.0
     (/
      (fma
       -1.0
       (sqrt (* (/ F B_m) 2.0))
       (* -0.5 (* (sqrt t_0) (* C (sqrt 2.0)))))
      A)
     (* 0.5 (sqrt (* t_0 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = F / pow(B_m, 3.0);
	return -A * fma(-1.0, (fma(-1.0, sqrt(((F / B_m) * 2.0)), (-0.5 * (sqrt(t_0) * (C * sqrt(2.0))))) / A), (0.5 * sqrt((t_0 * 2.0))));
}

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(F / (B_m ^ 3.0))
	return Float64(Float64(-A) * fma(-1.0, Float64(fma(-1.0, sqrt(Float64(Float64(F / B_m) * 2.0)), Float64(-0.5 * Float64(sqrt(t_0) * Float64(C * sqrt(2.0))))) / A), Float64(0.5 * sqrt(Float64(t_0 * 2.0)))))
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[(F / N[Power[B$95$m, 3.0], $MachinePrecision]), $MachinePrecision]}, N[((-A) * N[(-1.0 * N[(N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(C * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)} \]
4. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   2. sqrt-unprod N/A

      \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   10. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

   13. lift-+.f64 18.0

       \[\leadsto \mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right) \]

5. Applied rewrites 18.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(\sqrt{2} \cdot \left(A + C\right)\right)\right)\right)} \]

6. Taylor expanded in A around -inf

   \[\leadsto -1 \cdot \color{blue}{\left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(A \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(A \cdot \left(-1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{A} + \color{blue}{\frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto -1 \cdot \left(A \cdot \mathsf{fma}\left(-1, \frac{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right) + \frac{-1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)}{\color{blue}{A}}, \frac{1}{2} \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \sqrt{2}\right)\right)\right) \]

8. Applied rewrites 16.6%

   \[\leadsto -1 \cdot \color{blue}{\left(A \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{\frac{F}{{B}^{3}} \cdot 2}\right)\right)} \]

9. Final simplification 16.6%

   \[\leadsto \left(-A\right) \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \sqrt{\frac{F}{B} \cdot 2}, -0.5 \cdot \left(\sqrt{\frac{F}{{B}^{3}}} \cdot \left(C \cdot \sqrt{2}\right)\right)\right)}{A}, 0.5 \cdot \sqrt{\frac{F}{{B}^{3}} \cdot 2}\right) \]
10. Add Preprocessing

Alternative 13: 2.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in F around -inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. sqrt-unprod N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 7.6%

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

6. Taylor expanded in A around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. pow2 N/A

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

   8. pow2 N/A

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

   9. lower-hypot.f64 15.8

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

8. Applied rewrites 15.8%

   \[\leadsto \frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :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))))