ABCF->ab-angle b

Percentage Accurate: 18.6% → 55.1%

Time: 14.3s

Alternatives: 14

Speedup: 14.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(t\_0 \cdot F\right)\\ t_2 := -t\_0\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(e^{0.25}\right)}^{\left(\log \left(-2 \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)\right) - \log \left(\frac{-1}{F}\right)\right)}\right)}^{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1 (* 2.0 (* t_0 F)))
        (t_2 (- t_0))
        (t_3
         (/
          (sqrt (* t_1 (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_2)))
   (if (<= t_3 -1e-138)
     (*
      (- (sqrt 2.0))
      (sqrt
       (*
        F
        (/ (- (+ C A) (hypot (- A C) B_m)) (fma -4.0 (* C A) (* B_m B_m))))))
     (if (<= t_3 INFINITY)
       (/ (sqrt (* t_1 (+ (fma (/ (* B_m B_m) C) -0.5 A) A))) t_2)
       (/
        (pow
         (pow
          (exp 0.25)
          (- (log (* -2.0 (- A (hypot A B_m)))) (log (/ -1.0 F))))
         2.0)
        (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = 2.0 * (t_0 * F);
	double t_2 = -t_0;
	double t_3 = sqrt((t_1 * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -1e-138) {
		tmp = -sqrt(2.0) * sqrt((F * (((C + A) - hypot((A - C), B_m)) / fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (fma(((B_m * B_m) / C), -0.5, A) + A))) / t_2;
	} else {
		tmp = pow(pow(exp(0.25), (log((-2.0 * (A - hypot(A, B_m)))) - log((-1.0 / F)))), 2.0) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(2.0 * Float64(t_0 * F))
	t_2 = Float64(-t_0)
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= -1e-138)
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))) / t_2);
	else
		tmp = Float64(((exp(0.25) ^ Float64(log(Float64(-2.0 * Float64(A - hypot(A, B_m)))) - log(Float64(-1.0 / F)))) ^ 2.0) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-138], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-2.0 * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000007e-138

   1. Initial program 35.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   if -1.00000000000000007e-138 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 22.6

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

   5. Applied rewrites 22.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 20.1%

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

   9. Applied rewrites 20.1%

      \[\leadsto \frac{{\left({\left(\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2\right)}^{0.25}\right)}^{2}}{-\color{blue}{B}} \]

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 26.7%

       \[\leadsto \frac{{\left({\left(e^{0.25}\right)}^{\left(\log \left(-2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right)}\right)}^{2}}{-B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(e^{0.25}\right)}^{\left(\log \left(-2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right) - \log \left(\frac{-1}{F}\right)\right)}\right)}^{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(t\_0 \cdot F\right)\\ t_2 := -t\_0\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(e^{0.25}\right)}^{\left(\log \left(-2 \cdot F\right) + \log B\_m\right)}\right)}^{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1 (* 2.0 (* t_0 F)))
        (t_2 (- t_0))
        (t_3
         (/
          (sqrt (* t_1 (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_2)))
   (if (<= t_3 -1e-138)
     (*
      (- (sqrt 2.0))
      (sqrt
       (*
        F
        (/ (- (+ C A) (hypot (- A C) B_m)) (fma -4.0 (* C A) (* B_m B_m))))))
     (if (<= t_3 INFINITY)
       (/ (sqrt (* t_1 (+ (fma (/ (* B_m B_m) C) -0.5 A) A))) t_2)
       (/
        (pow (pow (exp 0.25) (+ (log (* -2.0 F)) (log B_m))) 2.0)
        (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = 2.0 * (t_0 * F);
	double t_2 = -t_0;
	double t_3 = sqrt((t_1 * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -1e-138) {
		tmp = -sqrt(2.0) * sqrt((F * (((C + A) - hypot((A - C), B_m)) / fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (fma(((B_m * B_m) / C), -0.5, A) + A))) / t_2;
	} else {
		tmp = pow(pow(exp(0.25), (log((-2.0 * F)) + log(B_m))), 2.0) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(2.0 * Float64(t_0 * F))
	t_2 = Float64(-t_0)
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= -1e-138)
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))) / t_2);
	else
		tmp = Float64(((exp(0.25) ^ Float64(log(Float64(-2.0 * F)) + log(B_m))) ^ 2.0) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-138], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Power[N[Power[N[Exp[0.25], $MachinePrecision], N[(N[Log[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000007e-138

   1. Initial program 35.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   if -1.00000000000000007e-138 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 22.6

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

   5. Applied rewrites 22.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 20.1%

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

   9. Applied rewrites 20.1%

      \[\leadsto \frac{{\left({\left(\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2\right)}^{0.25}\right)}^{2}}{-\color{blue}{B}} \]

   10. Taylor expanded in B around inf

       \[\leadsto \frac{{\left(e^{\frac{1}{4} \cdot \left(\log \left(-2 \cdot F\right) + -1 \cdot \log \left(\frac{1}{B}\right)\right)}\right)}^{2}}{-B} \]
   11. Step-by-step derivation

   12. Applied rewrites 23.1%

       \[\leadsto \frac{{\left({\left(e^{0.25}\right)}^{\left(\log \left(-2 \cdot F\right) + \left(-\left(-\log B\right)\right)\right)}\right)}^{2}}{-B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(e^{0.25}\right)}^{\left(\log \left(-2 \cdot F\right) + \log B\right)}\right)}^{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(t\_0 \cdot F\right)\\ t_2 := -t\_0\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1 (* 2.0 (* t_0 F)))
        (t_2 (- t_0))
        (t_3
         (/
          (sqrt (* t_1 (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_2)))
   (if (<= t_3 -1e-138)
     (*
      (- (sqrt 2.0))
      (sqrt
       (*
        F
        (/ (- (+ C A) (hypot (- A C) B_m)) (fma -4.0 (* C A) (* B_m B_m))))))
     (if (<= t_3 INFINITY)
       (/ (sqrt (* t_1 (+ (fma (/ (* B_m B_m) C) -0.5 A) A))) t_2)
       (/ (sqrt (* (* (- A (hypot A B_m)) F) 2.0)) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = 2.0 * (t_0 * F);
	double t_2 = -t_0;
	double t_3 = sqrt((t_1 * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -1e-138) {
		tmp = -sqrt(2.0) * sqrt((F * (((C + A) - hypot((A - C), B_m)) / fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (fma(((B_m * B_m) / C), -0.5, A) + A))) / t_2;
	} else {
		tmp = sqrt((((A - hypot(A, B_m)) * F) * 2.0)) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(2.0 * Float64(t_0 * F))
	t_2 = Float64(-t_0)
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= -1e-138)
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))) / t_2);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(A, B_m)) * F) * 2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-138], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000007e-138

   1. Initial program 35.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   if -1.00000000000000007e-138 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 22.6

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

   5. Applied rewrites 22.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 20.1%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ t_2 := -\sqrt{2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-138}:\\ \;\;\;\;t\_2 \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot F\right)} \cdot t\_2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0)))
        (t_2 (- (sqrt 2.0))))
   (if (<= t_1 -1e-138)
     (*
      t_2
      (sqrt
       (*
        F
        (/ (- (+ C A) (hypot (- A C) B_m)) (fma -4.0 (* C A) (* B_m B_m))))))
     (if (<= t_1 INFINITY)
       (/
        (*
         (sqrt
          (*
           (+ (fma -0.5 (/ (* B_m B_m) C) A) A)
           (* (fma (* -4.0 C) A (* B_m B_m)) F)))
         t_2)
        t_0)
       (/ (sqrt (* (* (- A (hypot A B_m)) F) 2.0)) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double t_2 = -sqrt(2.0);
	double tmp;
	if (t_1 <= -1e-138) {
		tmp = t_2 * sqrt((F * (((C + A) - hypot((A - C), B_m)) / fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt(((fma(-0.5, ((B_m * B_m) / C), A) + A) * (fma((-4.0 * C), A, (B_m * B_m)) * F))) * t_2) / t_0;
	} else {
		tmp = sqrt((((A - hypot(A, B_m)) * F) * 2.0)) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	t_2 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (t_1 <= -1e-138)
		tmp = Float64(t_2 * sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A) * Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * F))) * t_2) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(A, B_m)) * F) * 2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[t$95$1, -1e-138], N[(t$95$2 * N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision] * N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000007e-138

   1. Initial program 35.7%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 76.2%

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

   if -1.00000000000000007e-138 < (/.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 17.9%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.1

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

   5. Applied rewrites 6.1%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. pow1/2 N/A

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

      9. lift-sqrt.f64 N/A

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

   7. Applied rewrites 6.1%

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

   8. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 22.4

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

   10. Applied rewrites 22.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 20.1%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F\right)} \cdot \left(-\sqrt{2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ t_2 := -\sqrt{2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-177}:\\ \;\;\;\;t\_2 \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot F\right)} \cdot t\_2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0)))
        (t_2 (- (sqrt 2.0))))
   (if (<= t_1 -2e-177)
     (*
      t_2
      (sqrt
       (*
        F
        (/ (- (+ C A) (hypot (- A C) B_m)) (fma -4.0 (* C A) (* B_m B_m))))))
     (if (<= t_1 INFINITY)
       (/ (* (sqrt (* (+ A A) (* (fma (* -4.0 C) A (* B_m B_m)) F))) t_2) t_0)
       (/ (sqrt (* (* (- A (hypot A B_m)) F) 2.0)) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double t_2 = -sqrt(2.0);
	double tmp;
	if (t_1 <= -2e-177) {
		tmp = t_2 * sqrt((F * (((C + A) - hypot((A - C), B_m)) / fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt(((A + A) * (fma((-4.0 * C), A, (B_m * B_m)) * F))) * t_2) / t_0;
	} else {
		tmp = sqrt((((A - hypot(A, B_m)) * F) * 2.0)) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	t_2 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (t_1 <= -2e-177)
		tmp = Float64(t_2 * sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(A + A) * Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * F))) * t_2) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(A, B_m)) * F) * 2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[t$95$1, -2e-177], N[(t$95$2 * N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-177

   1. Initial program 36.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 75.3%

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

   if -1.9999999999999999e-177 < (/.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 16.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.4

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

   5. Applied rewrites 4.4%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. pow1/2 N/A

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

      9. lift-sqrt.f64 N/A

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. lower--.f64 N/A

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

      3. lower-neg.f64 21.0

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

   10. Applied rewrites 21.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 20.1%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-177}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F\right)} \cdot \left(-\sqrt{2}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-177}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\frac{-0.5}{A} \cdot B\_m\right) \cdot \frac{B\_m}{A} - 2\right) \cdot \left(-A\right)\right) \cdot \left(\left(t\_0 \cdot F\right) \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_1))))
   (if (<= t_2 -2e-177)
     (*
      (- (sqrt 2.0))
      (sqrt
       (*
        F
        (/ (- (+ C A) (hypot (- A C) B_m)) (fma -4.0 (* C A) (* B_m B_m))))))
     (if (<= t_2 INFINITY)
       (/
        (sqrt
         (*
          (* (- (* (* (/ -0.5 A) B_m) (/ B_m A)) 2.0) (- A))
          (* (* t_0 F) 2.0)))
        (- t_0))
       (/ (sqrt (* (* (- A (hypot A B_m)) F) 2.0)) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	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 tmp;
	if (t_2 <= -2e-177) {
		tmp = -sqrt(2.0) * sqrt((F * (((C + A) - hypot((A - C), B_m)) / fma(-4.0, (C * A), (B_m * B_m)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((((((-0.5 / A) * B_m) * (B_m / A)) - 2.0) * -A) * ((t_0 * F) * 2.0))) / -t_0;
	} else {
		tmp = sqrt((((A - hypot(A, B_m)) * F) * 2.0)) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = 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)))))) / Float64(-t_1))
	tmp = 0.0
	if (t_2 <= -2e-177)
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / fma(-4.0, Float64(C * A), Float64(B_m * B_m))))));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(-0.5 / A) * B_m) * Float64(B_m / A)) - 2.0) * Float64(-A)) * Float64(Float64(t_0 * F) * 2.0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(A, B_m)) * F) * 2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -2e-177], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(-0.5 / A), $MachinePrecision] * B$95$m), $MachinePrecision] * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * (-A)), $MachinePrecision] * N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-177

   1. Initial program 36.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 75.3%

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

   if -1.9999999999999999e-177 < (/.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 16.5%

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

   3. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 19.4

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

   5. Applied rewrites 19.4%

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

   7. Applied rewrites 19.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 20.1%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{-177}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\frac{-0.5}{A} \cdot B\right) \cdot \frac{B}{A} - 2\right) \cdot \left(-A\right)\right) \cdot \left(\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\frac{-0.5}{A} \cdot B\_m\right) \cdot \frac{B\_m}{A} - 2\right) \cdot \left(-A\right)\right) \cdot \left(\left(t\_0 \cdot F\right) \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, B\_m \cdot F, 2 \cdot \left(A \cdot F\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e-56)
     (/
      (sqrt
       (*
        (* (- (* (* (/ -0.5 A) B_m) (/ B_m A)) 2.0) (- A))
        (* (* t_0 F) 2.0)))
      (- t_0))
     (/ (sqrt (fma -2.0 (* B_m F) (* 2.0 (* A F)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-56) {
		tmp = sqrt(((((((-0.5 / A) * B_m) * (B_m / A)) - 2.0) * -A) * ((t_0 * F) * 2.0))) / -t_0;
	} else {
		tmp = sqrt(fma(-2.0, (B_m * F), (2.0 * (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-56)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(-0.5 / A) * B_m) * Float64(B_m / A)) - 2.0) * Float64(-A)) * Float64(Float64(t_0 * F) * 2.0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(B_m * F), Float64(2.0 * Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-56], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(-0.5 / A), $MachinePrecision] * B$95$m), $MachinePrecision] * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * (-A)), $MachinePrecision] * N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-56

   1. Initial program 19.9%

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

   3. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 16.0

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

   5. Applied rewrites 16.0%

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

   7. Applied rewrites 16.0%

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

   if 2.0000000000000001e-56 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.7%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 22.5

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

   5. Applied rewrites 22.5%

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

   7. Applied rewrites 22.6%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 17.5%

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

Alternative 8: 44.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 2.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\frac{-0.5}{A} \cdot B\_m\right) \cdot \frac{B\_m}{A} - 2\right) \cdot \left(-A\right)\right) \cdot \left(\left(t\_0 \cdot F\right) \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 2.2e-28)
     (/
      (sqrt
       (*
        (* (- (* (* (/ -0.5 A) B_m) (/ B_m A)) 2.0) (- A))
        (* (* t_0 F) 2.0)))
      (- t_0))
     (/ (sqrt (* (* (- A (hypot A B_m)) F) 2.0)) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 2.2e-28) {
		tmp = sqrt(((((((-0.5 / A) * B_m) * (B_m / A)) - 2.0) * -A) * ((t_0 * F) * 2.0))) / -t_0;
	} else {
		tmp = sqrt((((A - hypot(A, B_m)) * F) * 2.0)) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 2.2e-28)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(-0.5 / A) * B_m) * Float64(B_m / A)) - 2.0) * Float64(-A)) * Float64(Float64(t_0 * F) * 2.0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(A, B_m)) * F) * 2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.2e-28], N[(N[Sqrt[N[(N[(N[(N[(N[(N[(-0.5 / A), $MachinePrecision] * B$95$m), $MachinePrecision] * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision] * (-A)), $MachinePrecision] * N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.19999999999999996e-28

   1. Initial program 16.4%

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

   3. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 11.1

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

   5. Applied rewrites 11.1%

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

   7. Applied rewrites 11.1%

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

   if 2.19999999999999996e-28 < B

   1. Initial program 9.3%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 45.4

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

   5. Applied rewrites 45.4%

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

   7. Applied rewrites 45.5%

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

Alternative 9: 33.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, B\_m \cdot F, 2 \cdot \left(A \cdot F\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(A + A\right)\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5e-310)
   (/ (sqrt (fma -2.0 (* B_m F) (* 2.0 (* A F)))) (- B_m))
   (*
    (sqrt (* -8.0 (* (* A C) (+ A A))))
    (/ (- (sqrt F)) (fma (* -4.0 C) A (* B_m B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = sqrt(fma(-2.0, (B_m * F), (2.0 * (A * F)))) / -B_m;
	} else {
		tmp = sqrt((-8.0 * ((A * C) * (A + A)))) * (-sqrt(F) / fma((-4.0 * C), A, (B_m * B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(sqrt(fma(-2.0, Float64(B_m * F), Float64(2.0 * Float64(A * F)))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(A + A)))) * Float64(Float64(-sqrt(F)) / fma(Float64(-4.0 * C), A, Float64(B_m * B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5e-310], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 13.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 17.0

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

   5. Applied rewrites 17.0%

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

   7. Applied rewrites 17.1%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 12.5%

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

   if -4.999999999999985e-310 < F

   1. Initial program 21.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 42.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 42.5%

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

   7. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 24.4

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

   9. Applied rewrites 24.4%

      \[\leadsto \sqrt{\color{blue}{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)}} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, B \cdot F, 2 \cdot \left(A \cdot F\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(A + A\right)\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, B\_m \cdot F, 2 \cdot \left(A \cdot F\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot C\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5e-310)
   (/ (sqrt (fma -2.0 (* B_m F) (* 2.0 (* A F)))) (- B_m))
   (*
    (sqrt (* -16.0 (* (* A A) C)))
    (/ (- (sqrt F)) (fma (* -4.0 C) A (* B_m B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = sqrt(fma(-2.0, (B_m * F), (2.0 * (A * F)))) / -B_m;
	} else {
		tmp = sqrt((-16.0 * ((A * A) * C))) * (-sqrt(F) / fma((-4.0 * C), A, (B_m * B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(sqrt(fma(-2.0, Float64(B_m * F), Float64(2.0 * Float64(A * F)))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * A) * C))) * Float64(Float64(-sqrt(F)) / fma(Float64(-4.0 * C), A, Float64(B_m * B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5e-310], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) / N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 13.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 17.0

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

   5. Applied rewrites 17.0%

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

   7. Applied rewrites 17.1%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 12.5%

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

   if -4.999999999999985e-310 < F

   1. Initial program 21.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 42.6%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 42.5%

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

   7. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 15.2

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

   9. Applied rewrites 15.2%

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot A\right) \cdot C\right)}} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 29.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.62 \cdot 10^{+190}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, B\_m \cdot F, 2 \cdot \left(A \cdot F\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -1.62e+190)
   (/ (sqrt (* (* 4.0 A) F)) (- B_m))
   (/ (sqrt (fma -2.0 (* B_m F) (* 2.0 (* A F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -1.62e+190) {
		tmp = sqrt(((4.0 * A) * F)) / -B_m;
	} else {
		tmp = sqrt(fma(-2.0, (B_m * F), (2.0 * (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -1.62e+190)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * F)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(B_m * F), Float64(2.0 * Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -1.62e+190], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.62000000000000006e190

   1. Initial program 1.4%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 13.5

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

   5. Applied rewrites 13.5%

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

   7. Applied rewrites 13.5%

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

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot F\right)}}{-B} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.6%

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

   if -1.62000000000000006e190 < A

   1. Initial program 15.7%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 15.2

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

   5. Applied rewrites 15.2%

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

   7. Applied rewrites 15.2%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 11.9%

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

Alternative 12: 28.8% accurate, 12.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -6.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot F}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -6.8e+174)
   (/ (sqrt (* (* 4.0 A) F)) (- B_m))
   (/ (sqrt (* -2.0 (* B_m F))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -6.8e+174) {
		tmp = sqrt(((4.0 * A) * F)) / -B_m;
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-6.8d+174)) then
        tmp = sqrt(((4.0d0 * a) * f)) / -b_m
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / -b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -6.8e+174) {
		tmp = Math.sqrt(((4.0 * A) * F)) / -B_m;
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -6.8e+174:
		tmp = math.sqrt(((4.0 * A) * F)) / -B_m
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -6.8e+174)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * F)) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -6.8e+174)
		tmp = sqrt(((4.0 * A) * F)) / -B_m;
	else
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -6.8e+174], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -6.8000000000000002e174

   1. Initial program 1.3%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 12.0

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

   5. Applied rewrites 12.0%

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{4 \cdot \left(A \cdot F\right)}}{-B} \]
   9. Step-by-step derivation

   10. Applied rewrites 12.1%

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

   if -6.8000000000000002e174 < A

   1. Initial program 15.9%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 15.4

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

   5. Applied rewrites 15.4%

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

   7. Applied rewrites 15.4%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 12.9%

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

Alternative 13: 27.8% accurate, 14.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.4%

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

3. Taylor expanded in C around 0

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

   1. mul-1-neg N/A

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

   2. distribute-lft-neg-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. unpow2 N/A

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

   14. lower-hypot.f64 15.0

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

5. Applied rewrites 15.0%

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

7. Applied rewrites 15.0%

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

8. Taylor expanded in A around 0

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

10. Applied rewrites 11.8%

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

Alternative 14: 1.6% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.4%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 2.0

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

5. Applied rewrites 2.0%

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

7. Applied rewrites 2.0%

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

9. Applied rewrites 2.0%

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

Reproduce

herbie shell --seed 2024350 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))