ABCF->ab-angle a

Percentage Accurate: 19.5% → 51.8%

Time: 8.0s

Alternatives: 9

Speedup: 9.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 51.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := B\_m \cdot B\_m - t\_0\\ t_2 := {B\_m}^{2} - t\_0\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ t_4 := \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)} \cdot \sqrt{2 \cdot C}}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-199}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- (* B_m B_m) t_0))
        (t_2 (- (pow B_m 2.0) t_0))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_2))
        (t_4
         (/
          (-
           (*
            (sqrt (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F)))
            (sqrt (* 2.0 C))))
          t_1)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -2e-199)
       (/
        (-
         (sqrt
          (*
           (* 2.0 (* t_1 F))
           (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))))))
        t_1)
       (if (<= t_3 0.0)
         (* -1.0 (sqrt (* -1.0 (/ F A))))
         (if (<= t_3 INFINITY) t_4 (* -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 t_0 = (4.0 * A) * C;
	double t_1 = (B_m * B_m) - t_0;
	double t_2 = pow(B_m, 2.0) - t_0;
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double t_4 = -(sqrt((2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F))) * sqrt((2.0 * C))) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -2e-199) {
		tmp = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(Float64(B_m * B_m) - t_0)
	t_2 = Float64((B_m ^ 2.0) - t_0)
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_2)
	t_4 = Float64(Float64(-Float64(sqrt(Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F))) * sqrt(Float64(2.0 * C)))) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -2e-199)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))))) / t_1);
	elseif (t_3 <= 0.0)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[((-N[(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[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -2e-199], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 14.6%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 41.1

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

   4. Applied rewrites 41.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

   6. Applied rewrites 58.2%

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

   7. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 58.2

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

   9. Applied rewrites 58.2%

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

       1. lift-pow.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 58.2

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

   11. Applied rewrites 58.2%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.99999999999999996e-199

   1. Initial program 97.8%

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

   3. Applied rewrites 97.8%

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

   if -1.99999999999999996e-199 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 6.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 16.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 48.7

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

   7. Applied rewrites 48.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   3. 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 32.2

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

   4. Applied rewrites 32.2%

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

Alternative 2: 50.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)}\\ t_2 := B\_m \cdot B\_m - t\_0\\ t_3 := {B\_m}^{2} - t\_0\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{-t\_1 \cdot \sqrt{2 \cdot C}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-199}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{-t\_1 \cdot \sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (sqrt (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F))))
        (t_2 (- (* B_m B_m) t_0))
        (t_3 (- (pow B_m 2.0) t_0))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (- (* t_1 (sqrt (* 2.0 C)))) t_2)
     (if (<= t_4 -2e-199)
       (/
        (-
         (sqrt
          (*
           (* 2.0 (* t_2 F))
           (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))))))
        t_2)
       (if (<= t_4 INFINITY)
         (/ (- (* t_1 (sqrt (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))))) t_3)
         (* -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 t_0 = (4.0 * A) * C;
	double t_1 = sqrt((2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F)));
	double t_2 = (B_m * B_m) - t_0;
	double t_3 = pow(B_m, 2.0) - t_0;
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -(t_1 * sqrt((2.0 * C))) / t_2;
	} else if (t_4 <= -2e-199) {
		tmp = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = -(t_1 * sqrt(fma(-0.5, ((B_m * B_m) / A), (2.0 * C)))) / t_3;
	} else {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = sqrt(Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F)))
	t_2 = Float64(Float64(B_m * B_m) - t_0)
	t_3 = Float64((B_m ^ 2.0) - t_0)
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(-Float64(t_1 * sqrt(Float64(2.0 * C)))) / t_2);
	elseif (t_4 <= -2e-199)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(-Float64(t_1 * sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))))) / t_3);
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[((-N[(t$95$1 * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -2e-199], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[((-N[(t$95$1 * N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$3), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 31.5

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

   4. Applied rewrites 31.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

   6. Applied rewrites 47.7%

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

   7. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 47.7

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

   9. Applied rewrites 47.7%

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

       1. lift-pow.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 47.7

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

   11. Applied rewrites 47.7%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.99999999999999996e-199

   1. Initial program 97.8%

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

   3. Applied rewrites 97.8%

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

   if -1.99999999999999996e-199 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 19.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 48.3

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

   9. Applied rewrites 48.3%

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

   10. Taylor expanded in A around -inf

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       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 N/A

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

       5. lift-*.f64 57.1

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

   12. Applied rewrites 57.1%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   3. 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 32.2

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

   4. Applied rewrites 32.2%

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

Alternative 3: 50.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := {B\_m}^{2} - t\_0\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)} \cdot \sqrt{2 \cdot C}}{B\_m \cdot B\_m - t\_0}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+168}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-192}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- (pow B_m 2.0) t_0))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3
         (/
          (-
           (*
            (sqrt (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F)))
            (sqrt (* 2.0 C))))
          (- (* B_m B_m) t_0))))
   (if (<= t_2 -5e+168)
     t_3
     (if (<= t_2 -5e-192)
       (*
        -1.0
        (sqrt
         (*
          (/
           (* F (+ A (+ C (sqrt (fma B_m B_m (* (- A C) (- A C)))))))
           (- (* B_m B_m) (* 4.0 (* A C))))
          2.0)))
       (if (<= t_2 0.0)
         (* -1.0 (sqrt (* -1.0 (/ F A))))
         (if (<= t_2 INFINITY) t_3 (* -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 t_0 = (4.0 * A) * C;
	double t_1 = pow(B_m, 2.0) - t_0;
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = -(sqrt((2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F))) * sqrt((2.0 * C))) / ((B_m * B_m) - t_0);
	double tmp;
	if (t_2 <= -5e+168) {
		tmp = t_3;
	} else if (t_2 <= -5e-192) {
		tmp = -1.0 * sqrt((((F * (A + (C + sqrt(fma(B_m, B_m, ((A - C) * (A - C))))))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_2 <= 0.0) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64((B_m ^ 2.0) - t_0)
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = Float64(Float64(-Float64(sqrt(Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F))) * sqrt(Float64(2.0 * C)))) / Float64(Float64(B_m * B_m) - t_0))
	tmp = 0.0
	if (t_2 <= -5e+168)
		tmp = t_3;
	elseif (t_2 <= -5e-192)
		tmp = Float64(-1.0 * sqrt(Float64(Float64(Float64(F * Float64(A + Float64(C + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C))))))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C)))) * 2.0)));
	elseif (t_2 <= 0.0)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = 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[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+168], t$95$3, If[LessEqual[t$95$2, -5e-192], N[(-1.0 * N[Sqrt[N[(N[(N[(F * N[(A + N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.99999999999999967e168 or -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 17.3%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 42.2

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

   4. Applied rewrites 42.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

   6. Applied rewrites 58.8%

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

   7. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 58.8

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

   9. Applied rewrites 58.8%

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

       1. lift-pow.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 58.8

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

   11. Applied rewrites 58.8%

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

   if -4.99999999999999967e168 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000001e-192

   1. Initial program 97.8%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 88.6%

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

   if -5.0000000000000001e-192 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 6.8%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 15.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 48.4

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

   7. Applied rewrites 48.4%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   3. 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 32.2

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

   4. Applied rewrites 32.2%

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

Alternative 4: 48.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := {B\_m}^{2} - t\_0\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)} \cdot \sqrt{2 \cdot C}}{B\_m \cdot B\_m - t\_0}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+136}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-199}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- (pow B_m 2.0) t_0))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3
         (/
          (-
           (*
            (sqrt (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F)))
            (sqrt (* 2.0 C))))
          (- (* B_m B_m) t_0))))
   (if (<= t_2 -1e+136)
     t_3
     (if (<= t_2 -2e-199)
       (*
        -1.0
        (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))))
       (if (<= t_2 0.0)
         (* -1.0 (sqrt (* -1.0 (/ F A))))
         (if (<= t_2 INFINITY) t_3 (* -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 t_0 = (4.0 * A) * C;
	double t_1 = pow(B_m, 2.0) - t_0;
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = -(sqrt((2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F))) * sqrt((2.0 * C))) / ((B_m * B_m) - t_0);
	double tmp;
	if (t_2 <= -1e+136) {
		tmp = t_3;
	} else if (t_2 <= -2e-199) {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))));
	} else if (t_2 <= 0.0) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64((B_m ^ 2.0) - t_0)
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = Float64(Float64(-Float64(sqrt(Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F))) * sqrt(Float64(2.0 * C)))) / Float64(Float64(B_m * B_m) - t_0))
	tmp = 0.0
	if (t_2 <= -1e+136)
		tmp = t_3;
	elseif (t_2 <= -2e-199)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))));
	elseif (t_2 <= 0.0)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = 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[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+136], t$95$3, If[LessEqual[t$95$2, -2e-199], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * 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], If[LessEqual[t$95$2, 0.0], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000006e136 or -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 19.7%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 42.5

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

   4. Applied rewrites 42.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

   6. Applied rewrites 58.6%

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

   7. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 58.6

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

   9. Applied rewrites 58.6%

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

       1. lift-pow.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 58.6

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

   11. Applied rewrites 58.6%

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

   if -1.00000000000000006e136 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.99999999999999996e-199

   1. Initial program 97.8%

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

   2. Taylor expanded in A around 0

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

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

   if -1.99999999999999996e-199 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 6.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 16.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 48.7

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

   7. Applied rewrites 48.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   3. 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 32.2

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

   4. Applied rewrites 32.2%

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

Alternative 5: 43.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{if}\;B\_m \leq 7 \cdot 10^{-224}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B\_m \leq 1.15 \cdot 10^{-38}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 15500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B\_m \leq 2.05 \cdot 10^{+150}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* -1.0 (sqrt (* -1.0 (/ F A))))))
   (if (<= B_m 7e-224)
     t_0
     (if (<= B_m 1.15e-38)
       (/
        (- (sqrt (* (* 2.0 (* (fma -4.0 (* A C) (* B_m B_m)) F)) (* 2.0 C))))
        (- (* B_m B_m) (* (* 4.0 A) C)))
       (if (<= B_m 15500000000000.0)
         t_0
         (if (<= B_m 2.05e+150)
           (*
            -1.0
            (*
             (/ (sqrt 2.0) B_m)
             (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))))
           (* -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 t_0 = -1.0 * sqrt((-1.0 * (F / A)));
	double tmp;
	if (B_m <= 7e-224) {
		tmp = t_0;
	} else if (B_m <= 1.15e-38) {
		tmp = -sqrt(((2.0 * (fma(-4.0, (A * C), (B_m * B_m)) * F)) * (2.0 * C))) / ((B_m * B_m) - ((4.0 * A) * C));
	} else if (B_m <= 15500000000000.0) {
		tmp = t_0;
	} else if (B_m <= 2.05e+150) {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))));
	} else {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))))
	tmp = 0.0
	if (B_m <= 7e-224)
		tmp = t_0;
	elseif (B_m <= 1.15e-38)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * F)) * Float64(2.0 * C)))) / Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)));
	elseif (B_m <= 15500000000000.0)
		tmp = t_0;
	elseif (B_m <= 2.05e+150)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))));
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 7e-224], t$95$0, If[LessEqual[B$95$m, 1.15e-38], 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[(2.0 * C), $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, 15500000000000.0], t$95$0, If[LessEqual[B$95$m, 2.05e+150], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * 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[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 7.00000000000000037e-224 or 1.15000000000000001e-38 < B < 1.55e13

   1. Initial program 25.7%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 20.2%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if 7.00000000000000037e-224 < B < 1.15000000000000001e-38

   1. Initial program 22.9%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 45.0

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

   4. Applied rewrites 45.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

   6. Applied rewrites 44.5%

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

   7. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 44.5

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

   9. Applied rewrites 44.5%

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

   11. Applied rewrites 45.0%

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

   if 1.55e13 < B < 2.04999999999999997e150

   1. Initial program 30.4%

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

   2. Taylor expanded in A around 0

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

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 40.9

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

   4. Applied rewrites 40.9%

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

   if 2.04999999999999997e150 < B

   1. Initial program 0.3%

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

   2. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

Alternative 6: 41.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+23}:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 2e+23)
   (* -1.0 (sqrt (* -1.0 (/ F A))))
   (if (<= (pow B_m 2.0) 2e+292)
     (*
      -1.0
      (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))))
     (* -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 (pow(B_m, 2.0) <= 2e+23) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else if (pow(B_m, 2.0) <= 2e+292) {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))));
	} else {
		tmp = -1.0 * 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 ((B_m ^ 2.0) <= 2e+23)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	elseif ((B_m ^ 2.0) <= 2e+292)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))));
	else
		tmp = Float64(-1.0 * sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+23], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+292], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * 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[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999998e23

   1. Initial program 24.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.5

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

   7. Applied rewrites 36.5%

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

   if 1.9999999999999998e23 < (pow.f64 B #s(literal 2 binary64)) < 2e292

   1. Initial program 30.7%

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

   2. Taylor expanded in A around 0

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

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 40.4

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

   4. Applied rewrites 40.4%

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

   if 2e292 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.9%

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

   2. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   3. 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 51.8

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

   4. Applied rewrites 51.8%

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

Alternative 7: 39.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+193}:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 2e+193)
   (* -1.0 (sqrt (* -1.0 (/ F A))))
   (* -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 (pow(B_m, 2.0) <= 2e+193) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} 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 ((b_m ** 2.0d0) <= 2d+193) then
        tmp = (-1.0d0) * sqrt(((-1.0d0) * (f / a)))
    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 (Math.pow(B_m, 2.0) <= 2e+193) {
		tmp = -1.0 * Math.sqrt((-1.0 * (F / A)));
	} 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 math.pow(B_m, 2.0) <= 2e+193:
		tmp = -1.0 * math.sqrt((-1.0 * (F / A)))
	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 ((B_m ^ 2.0) <= 2e+193)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	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 ((B_m ^ 2.0) <= 2e+193)
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	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[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+193], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $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 (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000013e193

   1. Initial program 26.3%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 22.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.3

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

   7. Applied rewrites 34.3%

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

   if 2.00000000000000013e193 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.7%

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

   2. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   3. 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 49.3

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

   4. Applied rewrites 49.3%

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

Alternative 8: 27.2% accurate, 9.0× speedup

Math

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

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 (* -1.0 (/ F A)))))

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((-1.0 * (F / A)));
}

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(((-1.0d0) * (f / a)))
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((-1.0 * (F / A)));
}

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((-1.0 * (F / A)))

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(-1.0 * Float64(F / A))))
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((-1.0 * (F / A)));
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[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

2. Taylor expanded in F around 0

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

   1. lower-*.f64 N/A

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

   2. sqrt-unprod N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

4. Applied rewrites 18.3%

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

5. Taylor expanded in A around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 27.2

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

7. Applied rewrites 27.2%

   \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]
8. Add Preprocessing

Alternative 9: 2.4% accurate, 12.0× 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 19.5%

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

2. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
3. 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 28.1

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

4. Applied rewrites 28.1%

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

5. Taylor expanded in F around -inf

   \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{-1}\right)} \]
6. 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

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

7. Applied rewrites 2.4%

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

Reproduce

herbie shell --seed 2025113 
(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))))