ABCF->ab-angle a

Percentage Accurate: 18.5% → 56.3%

Time: 10.1s

Alternatives: 16

Speedup: 15.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.5% 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: 56.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-0.5, B\_m \cdot B\_m, 2 \cdot \left(A \cdot C\right)\right)}{A}\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 1.12 \cdot 10^{-166}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)} \cdot \sqrt{t\_0}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 8.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot t\_0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (fma -0.5 (* B_m B_m) (* 2.0 (* A C))) A))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= B_m 1.12e-166)
     (/
      (- (* (sqrt (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F))) (sqrt t_0)))
      t_1)
     (if (<= B_m 8.4e+26)
       (/ (- (sqrt (* (* 2.0 (* t_1 F)) t_0))) t_1)
       (*
        -1.0
        (* (/ (sqrt 2.0) B_m) (* (sqrt F) (sqrt (+ 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) {
	double t_0 = fma(-0.5, (B_m * B_m), (2.0 * (A * C))) / A;
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 1.12e-166) {
		tmp = -(sqrt((2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F))) * sqrt(t_0)) / t_1;
	} else if (B_m <= 8.4e+26) {
		tmp = -sqrt(((2.0 * (t_1 * F)) * t_0)) / t_1;
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(fma(-0.5, Float64(B_m * B_m), Float64(2.0 * Float64(A * C))) / A)
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 1.12e-166)
		tmp = Float64(Float64(-Float64(sqrt(Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F))) * sqrt(t_0))) / t_1);
	elseif (B_m <= 8.4e+26)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * t_0))) / t_1);
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C))))));
	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[(-0.5 * N[(B$95$m * B$95$m), $MachinePrecision] + N[(2.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.12e-166], N[((-N[(N[Sqrt[N[(2.0 * N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 8.4e+26], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.11999999999999994e-166

   1. Initial program 16.0%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 42.7

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

   10. Applied rewrites 42.7%

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

   11. Taylor expanded in A around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 42.7

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

   13. Applied rewrites 42.7%

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

   if 1.11999999999999994e-166 < B < 8.4000000000000003e26

   1. Initial program 28.7%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 40.5

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

   5. Applied rewrites 40.5%

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

   6. Taylor expanded in A around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 40.5

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

   8. Applied rewrites 40.5%

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

   if 8.4000000000000003e26 < B

   1. Initial program 12.7%

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

   3. Taylor expanded in A around 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 51.5

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

   5. Applied rewrites 51.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 74.0

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

   7. Applied rewrites 74.0%

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

Alternative 2: 56.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := 2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 1.12 \cdot 10^{-166}:\\ \;\;\;\;\frac{-\sqrt{t\_0} \cdot \sqrt{\frac{\mathsf{fma}\left(-0.5, B\_m \cdot B\_m, 2 \cdot \left(A \cdot C\right)\right)}{A}}}{{B\_m}^{2} - t\_1}\\ \mathbf{elif}\;B\_m \leq 8.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{t\_0 \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{B\_m \cdot B\_m - t\_1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

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 (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F)))
        (t_1 (* (* 4.0 A) C)))
   (if (<= B_m 1.12e-166)
     (/
      (- (* (sqrt t_0) (sqrt (/ (fma -0.5 (* B_m B_m) (* 2.0 (* A C))) A))))
      (- (pow B_m 2.0) t_1))
     (if (<= B_m 8.4e+26)
       (/
        (- (sqrt (* t_0 (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))))
        (- (* B_m B_m) t_1))
       (*
        -1.0
        (* (/ (sqrt 2.0) B_m) (* (sqrt F) (sqrt (+ 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) {
	double t_0 = 2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F);
	double t_1 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 1.12e-166) {
		tmp = -(sqrt(t_0) * sqrt((fma(-0.5, (B_m * B_m), (2.0 * (A * C))) / A))) / (pow(B_m, 2.0) - t_1);
	} else if (B_m <= 8.4e+26) {
		tmp = -sqrt((t_0 * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / ((B_m * B_m) - t_1);
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F))
	t_1 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 1.12e-166)
		tmp = Float64(Float64(-Float64(sqrt(t_0) * sqrt(Float64(fma(-0.5, Float64(B_m * B_m), Float64(2.0 * Float64(A * C))) / A)))) / Float64((B_m ^ 2.0) - t_1));
	elseif (B_m <= 8.4e+26)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))))) / Float64(Float64(B_m * B_m) - t_1));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C))))));
	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[(2.0 * N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 1.12e-166], N[((-N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(B$95$m * B$95$m), $MachinePrecision] + N[(2.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8.4e+26], N[((-N[Sqrt[N[(t$95$0 * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.11999999999999994e-166

   1. Initial program 16.0%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 42.7

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

   10. Applied rewrites 42.7%

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

   11. Taylor expanded in A around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 42.7

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

   13. Applied rewrites 42.7%

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

   if 1.11999999999999994e-166 < B < 8.4000000000000003e26

   1. Initial program 28.7%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 40.5

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

   5. Applied rewrites 40.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 41.0%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   12. Applied rewrites 40.6%

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

   if 8.4000000000000003e26 < B

   1. Initial program 12.7%

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

   3. Taylor expanded in A around 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 51.5

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

   5. Applied rewrites 51.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 74.0

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

   7. Applied rewrites 74.0%

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

Alternative 3: 51.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)\\ t_1 := 2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)\\ t_2 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;B\_m \leq 1.12 \cdot 10^{-166}:\\ \;\;\;\;\frac{-\sqrt{t\_1} \cdot \sqrt{t\_0}}{-4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 1.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-\sqrt{t\_1 \cdot t\_0}}{B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 4 \cdot 10^{+119}:\\ \;\;\;\;-1 \cdot \left(t\_2 \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(t\_2 \cdot \left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(A, B\_m\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))
        (t_1 (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F)))
        (t_2 (/ (sqrt 2.0) B_m)))
   (if (<= B_m 1.12e-166)
     (/ (- (* (sqrt t_1) (sqrt t_0))) (* -4.0 (* A C)))
     (if (<= B_m 1.25e+27)
       (/ (- (sqrt (* t_1 t_0))) (- (* B_m B_m) (* (* 4.0 A) C)))
       (if (<= B_m 4e+119)
         (* -1.0 (* t_2 (sqrt (* F (+ C (hypot B_m C))))))
         (* -1.0 (* t_2 (* (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 = fma(-0.5, ((B_m * B_m) / A), (2.0 * C));
	double t_1 = 2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F);
	double t_2 = sqrt(2.0) / B_m;
	double tmp;
	if (B_m <= 1.12e-166) {
		tmp = -(sqrt(t_1) * sqrt(t_0)) / (-4.0 * (A * C));
	} else if (B_m <= 1.25e+27) {
		tmp = -sqrt((t_1 * t_0)) / ((B_m * B_m) - ((4.0 * A) * C));
	} else if (B_m <= 4e+119) {
		tmp = -1.0 * (t_2 * sqrt((F * (C + hypot(B_m, C)))));
	} else {
		tmp = -1.0 * (t_2 * (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 = fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))
	t_1 = Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F))
	t_2 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if (B_m <= 1.12e-166)
		tmp = Float64(Float64(-Float64(sqrt(t_1) * sqrt(t_0))) / Float64(-4.0 * Float64(A * C)));
	elseif (B_m <= 1.25e+27)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * t_0))) / Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)));
	elseif (B_m <= 4e+119)
		tmp = Float64(-1.0 * Float64(t_2 * sqrt(Float64(F * Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(-1.0 * Float64(t_2 * 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[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[B$95$m, 1.12e-166], N[((-N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]) / N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.25e+27], N[((-N[Sqrt[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]) / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4e+119], N[(-1.0 * N[(t$95$2 * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(t$95$2 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.11999999999999994e-166

   1. Initial program 16.0%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 42.7

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

   10. Applied rewrites 42.7%

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

   11. Taylor expanded in A around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 42.7

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

   13. Applied rewrites 42.7%

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

   if 1.11999999999999994e-166 < B < 1.24999999999999995e27

   1. Initial program 28.7%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 40.5

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

   5. Applied rewrites 40.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 41.0%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   12. Applied rewrites 40.5%

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

   if 1.24999999999999995e27 < B < 3.99999999999999978e119

   1. Initial program 30.1%

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

   3. Taylor expanded in A around 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 46.0

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

   5. Applied rewrites 46.0%

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

   if 3.99999999999999978e119 < B

   1. Initial program 3.8%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 48.5

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

   5. Applied rewrites 48.5%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}\right)} \]
   6. 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. lift-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 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 4: 56.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)\\ t_1 := 2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)\\ \mathbf{if}\;B\_m \leq 1.12 \cdot 10^{-166}:\\ \;\;\;\;\frac{-\sqrt{t\_1} \cdot \sqrt{t\_0}}{-4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 8.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{t\_1 \cdot t\_0}}{B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))
        (t_1 (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F))))
   (if (<= B_m 1.12e-166)
     (/ (- (* (sqrt t_1) (sqrt t_0))) (* -4.0 (* A C)))
     (if (<= B_m 8.4e+26)
       (/ (- (sqrt (* t_1 t_0))) (- (* B_m B_m) (* (* 4.0 A) C)))
       (*
        -1.0
        (* (/ (sqrt 2.0) B_m) (* (sqrt F) (sqrt (+ 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) {
	double t_0 = fma(-0.5, ((B_m * B_m) / A), (2.0 * C));
	double t_1 = 2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F);
	double tmp;
	if (B_m <= 1.12e-166) {
		tmp = -(sqrt(t_1) * sqrt(t_0)) / (-4.0 * (A * C));
	} else if (B_m <= 8.4e+26) {
		tmp = -sqrt((t_1 * t_0)) / ((B_m * B_m) - ((4.0 * A) * C));
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * (sqrt(F) * sqrt((C + hypot(B_m, C)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))
	t_1 = Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F))
	tmp = 0.0
	if (B_m <= 1.12e-166)
		tmp = Float64(Float64(-Float64(sqrt(t_1) * sqrt(t_0))) / Float64(-4.0 * Float64(A * C)));
	elseif (B_m <= 8.4e+26)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * t_0))) / Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C))))));
	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[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.12e-166], N[((-N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]) / N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8.4e+26], N[((-N[Sqrt[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]) / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.11999999999999994e-166

   1. Initial program 16.0%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 42.7

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

   10. Applied rewrites 42.7%

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

   11. Taylor expanded in A around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 42.7

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

   13. Applied rewrites 42.7%

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

   if 1.11999999999999994e-166 < B < 8.4000000000000003e26

   1. Initial program 28.7%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 40.5

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

   5. Applied rewrites 40.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 41.0%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   12. Applied rewrites 40.6%

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

   if 8.4000000000000003e26 < B

   1. Initial program 12.7%

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

   3. Taylor expanded in A around 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 51.5

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

   5. Applied rewrites 51.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 74.0

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

   7. Applied rewrites 74.0%

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

Alternative 5: 51.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)\\ t_1 := 2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)\\ t_2 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;B\_m \leq 1.12 \cdot 10^{-166}:\\ \;\;\;\;\frac{-\sqrt{t\_1} \cdot \sqrt{t\_0}}{-4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 1.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-\sqrt{t\_1 \cdot t\_0}}{B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 1.5 \cdot 10^{+113}:\\ \;\;\;\;-1 \cdot \left(t\_2 \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(t\_2 \cdot \left(\sqrt{F} \cdot \sqrt{C + B\_m}\right)\right)\\ \end{array} \end{array} \]

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 (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))
        (t_1 (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F)))
        (t_2 (/ (sqrt 2.0) B_m)))
   (if (<= B_m 1.12e-166)
     (/ (- (* (sqrt t_1) (sqrt t_0))) (* -4.0 (* A C)))
     (if (<= B_m 1.25e+27)
       (/ (- (sqrt (* t_1 t_0))) (- (* B_m B_m) (* (* 4.0 A) C)))
       (if (<= B_m 1.5e+113)
         (* -1.0 (* t_2 (sqrt (* F (+ C (hypot B_m C))))))
         (* -1.0 (* t_2 (* (sqrt F) (sqrt (+ C 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 = fma(-0.5, ((B_m * B_m) / A), (2.0 * C));
	double t_1 = 2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F);
	double t_2 = sqrt(2.0) / B_m;
	double tmp;
	if (B_m <= 1.12e-166) {
		tmp = -(sqrt(t_1) * sqrt(t_0)) / (-4.0 * (A * C));
	} else if (B_m <= 1.25e+27) {
		tmp = -sqrt((t_1 * t_0)) / ((B_m * B_m) - ((4.0 * A) * C));
	} else if (B_m <= 1.5e+113) {
		tmp = -1.0 * (t_2 * sqrt((F * (C + hypot(B_m, C)))));
	} else {
		tmp = -1.0 * (t_2 * (sqrt(F) * sqrt((C + 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 = fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))
	t_1 = Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F))
	t_2 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if (B_m <= 1.12e-166)
		tmp = Float64(Float64(-Float64(sqrt(t_1) * sqrt(t_0))) / Float64(-4.0 * Float64(A * C)));
	elseif (B_m <= 1.25e+27)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * t_0))) / Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)));
	elseif (B_m <= 1.5e+113)
		tmp = Float64(-1.0 * Float64(t_2 * sqrt(Float64(F * Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(-1.0 * Float64(t_2 * Float64(sqrt(F) * sqrt(Float64(C + 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[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[B$95$m, 1.12e-166], N[((-N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]) / N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.25e+27], N[((-N[Sqrt[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]) / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.5e+113], N[(-1.0 * N[(t$95$2 * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(t$95$2 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.11999999999999994e-166

   1. Initial program 16.0%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 42.7

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

   10. Applied rewrites 42.7%

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

   11. Taylor expanded in A around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 42.7

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

   13. Applied rewrites 42.7%

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

   if 1.11999999999999994e-166 < B < 1.24999999999999995e27

   1. Initial program 28.7%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 40.5

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

   5. Applied rewrites 40.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 41.0%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   12. Applied rewrites 40.5%

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

   if 1.24999999999999995e27 < B < 1.5e113

   1. Initial program 30.6%

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

   3. Taylor expanded in A around 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 44.7

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

   5. Applied rewrites 44.7%

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

   if 1.5e113 < B

   1. Initial program 4.5%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   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 54.7

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

   5. Applied rewrites 54.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 82.9

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

   7. Applied rewrites 82.9%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 73.5%

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

Alternative 6: 51.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)\\ t_1 := 2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)\\ t_2 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;B\_m \leq 1.12 \cdot 10^{-166}:\\ \;\;\;\;\frac{-\sqrt{t\_1} \cdot \sqrt{t\_0}}{-4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 1.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-\sqrt{t\_1 \cdot t\_0}}{B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 1.5 \cdot 10^{+113}:\\ \;\;\;\;-1 \cdot \left(t\_2 \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(t\_2 \cdot \left(\sqrt{F} \cdot \sqrt{C + B\_m}\right)\right)\\ \end{array} \end{array} \]

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 (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))
        (t_1 (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F)))
        (t_2 (/ (sqrt 2.0) B_m)))
   (if (<= B_m 1.12e-166)
     (/ (- (* (sqrt t_1) (sqrt t_0))) (* -4.0 (* A C)))
     (if (<= B_m 1.25e+27)
       (/ (- (sqrt (* t_1 t_0))) (- (* B_m B_m) (* (* 4.0 A) C)))
       (if (<= B_m 1.5e+113)
         (* -1.0 (* t_2 (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))))
         (* -1.0 (* t_2 (* (sqrt F) (sqrt (+ C 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 = fma(-0.5, ((B_m * B_m) / A), (2.0 * C));
	double t_1 = 2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F);
	double t_2 = sqrt(2.0) / B_m;
	double tmp;
	if (B_m <= 1.12e-166) {
		tmp = -(sqrt(t_1) * sqrt(t_0)) / (-4.0 * (A * C));
	} else if (B_m <= 1.25e+27) {
		tmp = -sqrt((t_1 * t_0)) / ((B_m * B_m) - ((4.0 * A) * C));
	} else if (B_m <= 1.5e+113) {
		tmp = -1.0 * (t_2 * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))));
	} else {
		tmp = -1.0 * (t_2 * (sqrt(F) * sqrt((C + 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 = fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))
	t_1 = Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F))
	t_2 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if (B_m <= 1.12e-166)
		tmp = Float64(Float64(-Float64(sqrt(t_1) * sqrt(t_0))) / Float64(-4.0 * Float64(A * C)));
	elseif (B_m <= 1.25e+27)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * t_0))) / Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)));
	elseif (B_m <= 1.5e+113)
		tmp = Float64(-1.0 * Float64(t_2 * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))));
	else
		tmp = Float64(-1.0 * Float64(t_2 * Float64(sqrt(F) * sqrt(Float64(C + 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[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[B$95$m, 1.12e-166], N[((-N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]) / N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.25e+27], N[((-N[Sqrt[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]) / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.5e+113], N[(-1.0 * N[(t$95$2 * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(t$95$2 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.11999999999999994e-166

   1. Initial program 16.0%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 42.7%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 42.7

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

   10. Applied rewrites 42.7%

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

   11. Taylor expanded in A around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 42.7

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

   13. Applied rewrites 42.7%

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

   if 1.11999999999999994e-166 < B < 1.24999999999999995e27

   1. Initial program 28.7%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 40.5

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

   5. Applied rewrites 40.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 41.0%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   12. Applied rewrites 40.5%

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

   if 1.24999999999999995e27 < B < 1.5e113

   1. Initial program 30.6%

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

   3. Taylor expanded in A around 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 44.7

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

   5. Applied rewrites 44.7%

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

      1. lift-hypot.f64 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 40.0

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

   7. Applied rewrites 40.0%

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

   if 1.5e113 < B

   1. Initial program 4.5%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   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 54.7

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

   5. Applied rewrites 54.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 82.9

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

   7. Applied rewrites 82.9%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 73.5%

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

Alternative 7: 51.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;B\_m \leq 1.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 1.5 \cdot 10^{+113}:\\ \;\;\;\;-1 \cdot \left(t\_0 \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(t\_0 \cdot \left(\sqrt{F} \cdot \sqrt{C + B\_m}\right)\right)\\ \end{array} \end{array} \]

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 (/ (sqrt 2.0) B_m)))
   (if (<= B_m 1.25e+27)
     (/
      (-
       (sqrt
        (*
         (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F))
         (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))))
      (- (* B_m B_m) (* (* 4.0 A) C)))
     (if (<= B_m 1.5e+113)
       (* -1.0 (* t_0 (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))))
       (* -1.0 (* t_0 (* (sqrt F) (sqrt (+ C 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 = sqrt(2.0) / B_m;
	double tmp;
	if (B_m <= 1.25e+27) {
		tmp = -sqrt(((2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F)) * fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / ((B_m * B_m) - ((4.0 * A) * C));
	} else if (B_m <= 1.5e+113) {
		tmp = -1.0 * (t_0 * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))));
	} else {
		tmp = -1.0 * (t_0 * (sqrt(F) * sqrt((C + 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(sqrt(2.0) / B_m)
	tmp = 0.0
	if (B_m <= 1.25e+27)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F)) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))))) / Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)));
	elseif (B_m <= 1.5e+113)
		tmp = Float64(-1.0 * Float64(t_0 * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))));
	else
		tmp = Float64(-1.0 * Float64(t_0 * Float64(sqrt(F) * sqrt(Float64(C + 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[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[B$95$m, 1.25e+27], N[((-N[Sqrt[N[(N[(2.0 * N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 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]) / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.5e+113], N[(-1.0 * N[(t$95$0 * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(t$95$0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.24999999999999995e27

   1. Initial program 23.5%

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

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 42.2

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

   5. Applied rewrites 42.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in A around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 41.7

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

   10. Applied rewrites 41.7%

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

   12. Applied rewrites 42.2%

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

   if 1.24999999999999995e27 < B < 1.5e113

   1. Initial program 30.6%

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

   3. Taylor expanded in A around 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 44.7

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

   5. Applied rewrites 44.7%

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

      1. lift-hypot.f64 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 40.0

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

   7. Applied rewrites 40.0%

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

   if 1.5e113 < B

   1. Initial program 4.5%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   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 54.7

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

   5. Applied rewrites 54.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 82.9

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

   7. Applied rewrites 82.9%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 73.5%

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

Alternative 8: 37.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;A \leq -3.8 \cdot 10^{+146}:\\ \;\;\;\;-1 \cdot \left(t\_0 \cdot \sqrt{-0.5 \cdot \frac{\left(B\_m \cdot B\_m\right) \cdot F}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(t\_0 \cdot \left(\sqrt{F} \cdot \sqrt{C + B\_m}\right)\right)\\ \end{array} \end{array} \]

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 (/ (sqrt 2.0) B_m)))
   (if (<= A -3.8e+146)
     (* -1.0 (* t_0 (sqrt (* -0.5 (/ (* (* B_m B_m) F) A)))))
     (* -1.0 (* t_0 (* (sqrt F) (sqrt (+ C 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 = sqrt(2.0) / B_m;
	double tmp;
	if (A <= -3.8e+146) {
		tmp = -1.0 * (t_0 * sqrt((-0.5 * (((B_m * B_m) * F) / A))));
	} else {
		tmp = -1.0 * (t_0 * (sqrt(F) * sqrt((C + B_m))));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(2.0d0) / b_m
    if (a <= (-3.8d+146)) then
        tmp = (-1.0d0) * (t_0 * sqrt(((-0.5d0) * (((b_m * b_m) * f) / a))))
    else
        tmp = (-1.0d0) * (t_0 * (sqrt(f) * sqrt((c + b_m))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double tmp;
	if (A <= -3.8e+146) {
		tmp = -1.0 * (t_0 * Math.sqrt((-0.5 * (((B_m * B_m) * F) / A))));
	} else {
		tmp = -1.0 * (t_0 * (Math.sqrt(F) * Math.sqrt((C + 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.sqrt(2.0) / B_m
	tmp = 0
	if A <= -3.8e+146:
		tmp = -1.0 * (t_0 * math.sqrt((-0.5 * (((B_m * B_m) * F) / A))))
	else:
		tmp = -1.0 * (t_0 * (math.sqrt(F) * math.sqrt((C + 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(sqrt(2.0) / B_m)
	tmp = 0.0
	if (A <= -3.8e+146)
		tmp = Float64(-1.0 * Float64(t_0 * sqrt(Float64(-0.5 * Float64(Float64(Float64(B_m * B_m) * F) / A)))));
	else
		tmp = Float64(-1.0 * Float64(t_0 * Float64(sqrt(F) * sqrt(Float64(C + 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 = sqrt(2.0) / B_m;
	tmp = 0.0;
	if (A <= -3.8e+146)
		tmp = -1.0 * (t_0 * sqrt((-0.5 * (((B_m * B_m) * F) / A))));
	else
		tmp = -1.0 * (t_0 * (sqrt(F) * sqrt((C + 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[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[A, -3.8e+146], N[(-1.0 * N[(t$95$0 * N[Sqrt[N[(-0.5 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(t$95$0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < -3.79999999999999979e146

   1. Initial program 0.8%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 12.1

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

   5. Applied rewrites 12.1%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 20.3

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

   8. Applied rewrites 20.3%

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

   if -3.79999999999999979e146 < A

   1. Initial program 24.3%

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

   3. Taylor expanded in 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 38.9

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

   5. Applied rewrites 38.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 52.2

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 43.5%

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

Alternative 9: 38.1% accurate, 7.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B\_m}\\ \mathbf{if}\;C \leq 2.7 \cdot 10^{+203}:\\ \;\;\;\;-1 \cdot \left(t\_0 \cdot \left(\sqrt{F} \cdot \sqrt{C + B\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(t\_0 \cdot \left(\sqrt{F} \cdot \sqrt{C + C}\right)\right)\\ \end{array} \end{array} \]

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 (/ (sqrt 2.0) B_m)))
   (if (<= C 2.7e+203)
     (* -1.0 (* t_0 (* (sqrt F) (sqrt (+ C B_m)))))
     (* -1.0 (* t_0 (* (sqrt F) (sqrt (+ C C))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / B_m;
	double tmp;
	if (C <= 2.7e+203) {
		tmp = -1.0 * (t_0 * (sqrt(F) * sqrt((C + B_m))));
	} else {
		tmp = -1.0 * (t_0 * (sqrt(F) * sqrt((C + C))));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(2.0d0) / b_m
    if (c <= 2.7d+203) then
        tmp = (-1.0d0) * (t_0 * (sqrt(f) * sqrt((c + b_m))))
    else
        tmp = (-1.0d0) * (t_0 * (sqrt(f) * sqrt((c + c))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double tmp;
	if (C <= 2.7e+203) {
		tmp = -1.0 * (t_0 * (Math.sqrt(F) * Math.sqrt((C + B_m))));
	} else {
		tmp = -1.0 * (t_0 * (Math.sqrt(F) * Math.sqrt((C + C))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	tmp = 0
	if C <= 2.7e+203:
		tmp = -1.0 * (t_0 * (math.sqrt(F) * math.sqrt((C + B_m))))
	else:
		tmp = -1.0 * (t_0 * (math.sqrt(F) * math.sqrt((C + C))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if (C <= 2.7e+203)
		tmp = Float64(-1.0 * Float64(t_0 * Float64(sqrt(F) * sqrt(Float64(C + B_m)))));
	else
		tmp = Float64(-1.0 * Float64(t_0 * Float64(sqrt(F) * sqrt(Float64(C + C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	tmp = 0.0;
	if (C <= 2.7e+203)
		tmp = -1.0 * (t_0 * (sqrt(F) * sqrt((C + B_m))));
	else
		tmp = -1.0 * (t_0 * (sqrt(F) * sqrt((C + C))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[C, 2.7e+203], N[(-1.0 * N[(t$95$0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(t$95$0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 2.7e203

   1. Initial program 21.9%

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

   3. Taylor expanded in A around 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 35.2

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

   5. Applied rewrites 35.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 46.0

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

   7. Applied rewrites 46.0%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 41.1%

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

   if 2.7e203 < C

   1. Initial program 1.6%

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

   3. Taylor expanded in A around 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 17.6

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

   5. Applied rewrites 17.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-prod N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-+.f64 30.4

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

   7. Applied rewrites 30.4%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 23.3%

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

Alternative 10: 36.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{C + B\_m}\right)\right) \end{array} \]

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
 (* -1.0 (* (/ (sqrt 2.0) B_m) (* (sqrt F) (sqrt (+ C B_m))))))

C

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

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-1.0d0) * ((sqrt(2.0d0) / b_m) * (sqrt(f) * sqrt((c + b_m))))
end function

Java

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

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -1.0 * ((math.sqrt(2.0) / B_m) * (math.sqrt(F) * math.sqrt((C + B_m))))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * sqrt(Float64(C + B_m)))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -1.0 * ((sqrt(2.0) / B_m) * (sqrt(F) * sqrt((C + B_m))));
end

Wolfram

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

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
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 32.2

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

5. Applied rewrites 32.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-hypot.f64 N/A

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

   5. sqrt-prod N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lift-hypot.f64 N/A

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

   14. lift-+.f64 43.3

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

7. Applied rewrites 43.3%

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

8. Taylor expanded in B around inf

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

10. Applied rewrites 36.5%

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

Alternative 11: 35.8% accurate, 8.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \sqrt{B\_m}\right)\right) \end{array} \]

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
 (* -1.0 (* (/ (sqrt 2.0) B_m) (* (sqrt F) (sqrt B_m)))))

C

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

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-1.0d0) * ((sqrt(2.0d0) / b_m) * (sqrt(f) * sqrt(b_m)))
end function

Java

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

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -1.0 * ((math.sqrt(2.0) / B_m) * (math.sqrt(F) * math.sqrt(B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * sqrt(B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -1.0 * ((sqrt(2.0) / B_m) * (sqrt(F) * sqrt(B_m)));
end

Wolfram

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

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
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 32.2

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

5. Applied rewrites 32.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-hypot.f64 N/A

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

   5. sqrt-prod N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lift-hypot.f64 N/A

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

   14. lift-+.f64 43.3

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

7. Applied rewrites 43.3%

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

8. Taylor expanded in B around inf

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

10. Applied rewrites 35.8%

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

Alternative 12: 34.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 4.1 \cdot 10^{-92}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + B\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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 4.1e-92)
   (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ C B_m)))))
   (* -1.0 (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 <= 4.1e-92) {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + B_m))));
	} else {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 4.1d-92) then
        tmp = (-1.0d0) * ((sqrt(2.0d0) / b_m) * sqrt((f * (c + b_m))))
    else
        tmp = (-1.0d0) * sqrt(((f / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 4.1e-92) {
		tmp = -1.0 * ((Math.sqrt(2.0) / B_m) * Math.sqrt((F * (C + B_m))));
	} else {
		tmp = -1.0 * 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 <= 4.1e-92:
		tmp = -1.0 * ((math.sqrt(2.0) / B_m) * math.sqrt((F * (C + B_m))))
	else:
		tmp = -1.0 * 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 <= 4.1e-92)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(C + B_m)))));
	else
		tmp = Float64(-1.0 * 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 <= 4.1e-92)
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + B_m))));
	else
		tmp = -1.0 * 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, 4.1e-92], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C + B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 4.1000000000000002e-92

   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 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 34.7

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 29.3%

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

   if 4.1000000000000002e-92 < F

   1. Initial program 15.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 38.2

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

   5. Applied rewrites 38.2%

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

Alternative 13: 34.2% accurate, 9.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 4.4 \cdot 10^{-93}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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 4.4e-93)
   (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F B_m))))
   (* -1.0 (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 <= 4.4e-93) {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * B_m)));
	} else {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 4.4d-93) then
        tmp = (-1.0d0) * ((sqrt(2.0d0) / b_m) * sqrt((f * b_m)))
    else
        tmp = (-1.0d0) * sqrt(((f / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 4.4e-93) {
		tmp = -1.0 * ((Math.sqrt(2.0) / B_m) * Math.sqrt((F * B_m)));
	} else {
		tmp = -1.0 * 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 <= 4.4e-93:
		tmp = -1.0 * ((math.sqrt(2.0) / B_m) * math.sqrt((F * B_m)))
	else:
		tmp = -1.0 * 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 <= 4.4e-93)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * B_m))));
	else
		tmp = Float64(-1.0 * 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 <= 4.4e-93)
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * B_m)));
	else
		tmp = -1.0 * 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, 4.4e-93], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 4.39999999999999991e-93

   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 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 34.6

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

   5. Applied rewrites 34.6%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 28.9%

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

   if 4.39999999999999991e-93 < F

   1. Initial program 15.9%

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

   3. Taylor expanded in 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 38.2

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

   5. Applied rewrites 38.2%

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

Alternative 14: 28.1% accurate, 11.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 8.8 \cdot 10^{+80}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{2}{B\_m} \cdot \sqrt{C \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 8.8e+80)
   (* -1.0 (sqrt (* (/ F B_m) 2.0)))
   (* -1.0 (* (/ 2.0 B_m) (sqrt (* C F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 8.8e+80) {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	} else {
		tmp = -1.0 * ((2.0 / B_m) * sqrt((C * F)));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 8.8d+80) then
        tmp = (-1.0d0) * sqrt(((f / b_m) * 2.0d0))
    else
        tmp = (-1.0d0) * ((2.0d0 / b_m) * sqrt((c * f)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 8.8e+80) {
		tmp = -1.0 * Math.sqrt(((F / B_m) * 2.0));
	} else {
		tmp = -1.0 * ((2.0 / B_m) * Math.sqrt((C * F)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 8.8e+80:
		tmp = -1.0 * math.sqrt(((F / B_m) * 2.0))
	else:
		tmp = -1.0 * ((2.0 / B_m) * math.sqrt((C * F)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 8.8e+80)
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	else
		tmp = Float64(-1.0 * Float64(Float64(2.0 / B_m) * sqrt(Float64(C * F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 8.8e+80)
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	else
		tmp = -1.0 * ((2.0 / B_m) * sqrt((C * F)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 8.8e+80], N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(2.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 8.80000000000000011e80

   1. Initial program 21.7%

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

   3. Taylor expanded in 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 34.8

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

   5. Applied rewrites 34.8%

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

   if 8.80000000000000011e80 < C

   1. Initial program 12.3%

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

   3. Taylor expanded in 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 23.8

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

   5. Applied rewrites 23.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-pow2 N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 14.6

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

   8. Applied rewrites 14.6%

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

Alternative 15: 27.7% accurate, 15.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2} \end{array} \]

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 (* -1.0 (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) {
	return -1.0 * sqrt(((F / B_m) * 2.0));
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-1.0d0) * sqrt(((f / b_m) * 2.0d0))
end function

Java

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

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 -1.0 * math.sqrt(((F / B_m) * 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)
	return Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)))
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 = -1.0 * sqrt(((F / B_m) * 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_] := N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.5%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 27.7

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

5. Applied rewrites 27.7%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
6. Add Preprocessing

Alternative 16: 2.4% accurate, 18.2× speedup

Math

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

FPCore

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

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt(((f / b_m) * 2.0d0))
end function

Java

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

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt(((F / B_m) * 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)
	return sqrt(Float64(Float64(F / B_m) * 2.0))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt(((F / B_m) * 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_] := N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 18.5%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 27.7

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

5. Applied rewrites 27.7%

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

6. Taylor expanded in F around -inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. sqrt-prod N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sqrt.f64 2.4

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

8. Applied rewrites 2.4%

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

Reproduce

herbie shell --seed 2025088 
(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))))