ABCF->ab-angle a

Percentage Accurate: 19.4% → 66.0%

Time: 17.5s

Alternatives: 17

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.4% 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: 66.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := -t\_2\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ t_5 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{t\_0}\right)\right) \cdot \frac{t\_1}{t\_5}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot t\_5} \cdot t\_1}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (sqrt (+ (+ (hypot (- A C) B_m) A) C)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3 (- t_2))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_3))
        (t_5 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_4 -4e-206)
     (* (* (sqrt (* F 2.0)) (- (sqrt t_0))) (/ t_1 t_5))
     (if (<= t_4 2e+216)
       (/
        (sqrt (* (* t_0 2.0) (* F (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)))))
        t_3)
       (if (<= t_4 INFINITY)
         (/ (* (sqrt (* (* 2.0 F) t_5)) t_1) t_3)
         (*
          (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot C B_m) C)))
          (- (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = sqrt(((hypot((A - C), B_m) + A) + C));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -t_2;
	double t_4 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double t_5 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_4 <= -4e-206) {
		tmp = (sqrt((F * 2.0)) * -sqrt(t_0)) * (t_1 / t_5);
	} else if (t_4 <= 2e+216) {
		tmp = sqrt(((t_0 * 2.0) * (F * fma(-0.5, ((B_m * B_m) / A), (C * 2.0))))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(((2.0 * F) * t_5)) * t_1) / t_3;
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(C, B_m) + C))) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(-t_2)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_3)
	t_5 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_4 <= -4e-206)
		tmp = Float64(Float64(sqrt(Float64(F * 2.0)) * Float64(-sqrt(t_0))) * Float64(t_1 / t_5));
	elseif (t_4 <= 2e+216)
		tmp = Float64(sqrt(Float64(Float64(t_0 * 2.0) * Float64(F * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * t_5)) * t_1) / t_3);
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(C, B_m) + C))) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e-206], N[(N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision] * N[(t$95$1 / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+216], N[(N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.00000000000000011e-206

   1. Initial program 51.1%

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

   4. Applied rewrites 71.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. pow1/2 N/A

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

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-pow.f64 N/A

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

      21. metadata-eval N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. associate-*l* N/A

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

      25. lift-*.f64 N/A

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

      26. lift-*.f64 N/A

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

   6. Applied rewrites 86.2%

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

   if -4.00000000000000011e-206 < (/.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))) < 2e216

   1. Initial program 22.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.2

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

   5. Applied rewrites 34.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 33.6%

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

   if 2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 85.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 16.4

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

   5. Applied rewrites 16.4%

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

   7. Applied rewrites 23.5%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}\\ t_5 := \left(2 \cdot F\right) \cdot t\_0\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4 \cdot \frac{\sqrt{F \cdot 2}}{-B\_m}\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot t\_5}}{\mathsf{fma}\left(-B\_m, B\_m, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right) \cdot \left(\left(t\_1 \cdot F\right) \cdot 2\right)}}{-t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_5} \cdot \frac{\sqrt{C \cdot 2}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot t\_4\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (sqrt (+ (hypot C B_m) C)))
        (t_5 (* (* 2.0 F) t_0)))
   (if (<= t_3 (- INFINITY))
     (* t_4 (/ (sqrt (* F 2.0)) (- B_m)))
     (if (<= t_3 -4e-206)
       (/
        (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) t_5))
        (fma (- B_m) B_m (* C (* A 4.0))))
       (if (<= t_3 2e+216)
         (/
          (sqrt (* (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)) (* (* t_1 F) 2.0)))
          (- t_1))
         (if (<= t_3 INFINITY)
           (* (sqrt t_5) (/ (sqrt (* C 2.0)) (- t_0)))
           (* (* (/ (sqrt 2.0) B_m) t_4) (- (sqrt F)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = sqrt((hypot(C, B_m) + C));
	double t_5 = (2.0 * F) * t_0;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4 * (sqrt((F * 2.0)) / -B_m);
	} else if (t_3 <= -4e-206) {
		tmp = sqrt((((hypot((A - C), B_m) + A) + C) * t_5)) / fma(-B_m, B_m, (C * (A * 4.0)));
	} else if (t_3 <= 2e+216) {
		tmp = sqrt((fma(-0.5, ((B_m * B_m) / A), (C * 2.0)) * ((t_1 * F) * 2.0))) / -t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(t_5) * (sqrt((C * 2.0)) / -t_0);
	} else {
		tmp = ((sqrt(2.0) / B_m) * t_4) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = sqrt(Float64(hypot(C, B_m) + C))
	t_5 = Float64(Float64(2.0 * F) * t_0)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(t_4 * Float64(sqrt(Float64(F * 2.0)) / Float64(-B_m)));
	elseif (t_3 <= -4e-206)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * t_5)) / fma(Float64(-B_m), B_m, Float64(C * Float64(A * 4.0))));
	elseif (t_3 <= 2e+216)
		tmp = Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)) * Float64(Float64(t_1 * F) * 2.0))) / Float64(-t_1));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(t_5) * Float64(sqrt(Float64(C * 2.0)) / Float64(-t_0)));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * t_4) * Float64(-sqrt(F)));
	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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(t$95$4 * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -4e-206], N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / N[((-B$95$m) * B$95$m + N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+216], N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * F), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[t$95$5], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * t$95$4), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 21.4

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

   5. Applied rewrites 21.4%

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

   7. Applied rewrites 28.0%

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

   9. Applied rewrites 28.2%

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

   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))) < -4.00000000000000011e-206

   1. Initial program 96.6%

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

   4. Applied rewrites 96.6%

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

   if -4.00000000000000011e-206 < (/.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))) < 2e216

   1. Initial program 22.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.2

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

   5. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 34.2%

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

   if 2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.6%

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0 \cdot \frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 36.7

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + 0 \cdot \color{blue}{\frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 36.7%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(2 + 0 \cdot \frac{A}{C}\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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 A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 16.4

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

   5. Applied rewrites 16.4%

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

   7. Applied rewrites 23.5%

      \[\leadsto -\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \sqrt{F} \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.5%

   \[\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 -\infty:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \frac{\sqrt{F \cdot 2}}{-B}\\ \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 -4 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-B, B, C \cdot \left(A \cdot 4\right)\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 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, 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:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := -t\_1\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{t\_0}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot t\_4} \cdot \frac{\sqrt{C \cdot 2}}{-t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2 (- t_1))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_2))
        (t_4 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_3 -4e-206)
     (*
      (* (sqrt (* F 2.0)) (- (sqrt t_0)))
      (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) t_4))
     (if (<= t_3 2e+216)
       (/
        (sqrt (* (* t_0 2.0) (* F (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)))))
        t_2)
       (if (<= t_3 INFINITY)
         (* (sqrt (* (* 2.0 F) t_4)) (/ (sqrt (* C 2.0)) (- t_4)))
         (*
          (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot C B_m) C)))
          (- (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -t_1;
	double t_3 = sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double t_4 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_3 <= -4e-206) {
		tmp = (sqrt((F * 2.0)) * -sqrt(t_0)) * (sqrt(((hypot((A - C), B_m) + A) + C)) / t_4);
	} else if (t_3 <= 2e+216) {
		tmp = sqrt(((t_0 * 2.0) * (F * fma(-0.5, ((B_m * B_m) / A), (C * 2.0))))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * F) * t_4)) * (sqrt((C * 2.0)) / -t_4);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(C, B_m) + C))) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(-t_1)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_2)
	t_4 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_3 <= -4e-206)
		tmp = Float64(Float64(sqrt(Float64(F * 2.0)) * Float64(-sqrt(t_0))) * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / t_4));
	elseif (t_3 <= 2e+216)
		tmp = Float64(sqrt(Float64(Float64(t_0 * 2.0) * Float64(F * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * t_4)) * Float64(sqrt(Float64(C * 2.0)) / Float64(-t_4)));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(C, B_m) + C))) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + 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 = (-t$95$1)}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -4e-206], N[(N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$0], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+216], N[(N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.00000000000000011e-206

   1. Initial program 51.1%

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

   4. Applied rewrites 71.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. pow1/2 N/A

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

      15. lift-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. fp-cancel-sign-sub-inv N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-pow.f64 N/A

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

      21. metadata-eval N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. associate-*l* N/A

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

      25. lift-*.f64 N/A

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

      26. lift-*.f64 N/A

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

   6. Applied rewrites 86.2%

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

   if -4.00000000000000011e-206 < (/.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))) < 2e216

   1. Initial program 22.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.2

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

   5. Applied rewrites 34.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 33.6%

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

   if 2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.6%

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0 \cdot \frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 36.7

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + 0 \cdot \color{blue}{\frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 36.7%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(2 + 0 \cdot \frac{A}{C}\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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 A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 16.4

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

   5. Applied rewrites 16.4%

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

   7. Applied rewrites 23.5%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2\\ t_1 := -\sqrt{F}\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := -t\_2\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ t_5 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot t\_1\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_5}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot t\_5} \cdot \frac{\sqrt{C \cdot 2}}{-t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}\right) \cdot t\_1\\ \end{array} \end{array} \]

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 (* C A) -4.0 (* B_m B_m)) 2.0))
        (t_1 (- (sqrt F)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3 (- t_2))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_3))
        (t_5 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_4 -4e-206)
     (* (* (sqrt t_0) t_1) (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) t_5))
     (if (<= t_4 2e+216)
       (/ (sqrt (* t_0 (* F (fma -0.5 (/ (* B_m B_m) A) (* C 2.0))))) t_3)
       (if (<= t_4 INFINITY)
         (* (sqrt (* (* 2.0 F) t_5)) (/ (sqrt (* C 2.0)) (- t_5)))
         (* (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot C B_m) C))) t_1))))))

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((C * A), -4.0, (B_m * B_m)) * 2.0;
	double t_1 = -sqrt(F);
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -t_2;
	double t_4 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double t_5 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_4 <= -4e-206) {
		tmp = (sqrt(t_0) * t_1) * (sqrt(((hypot((A - C), B_m) + A) + C)) / t_5);
	} else if (t_4 <= 2e+216) {
		tmp = sqrt((t_0 * (F * fma(-0.5, ((B_m * B_m) / A), (C * 2.0))))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * F) * t_5)) * (sqrt((C * 2.0)) / -t_5);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(C, B_m) + C))) * t_1;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0)
	t_1 = Float64(-sqrt(F))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(-t_2)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_3)
	t_5 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_4 <= -4e-206)
		tmp = Float64(Float64(sqrt(t_0) * t_1) * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / t_5));
	elseif (t_4 <= 2e+216)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * t_5)) * Float64(sqrt(Float64(C * 2.0)) / Float64(-t_5)));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(C, B_m) + C))) * t_1);
	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[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e-206], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+216], N[(N[Sqrt[N[(t$95$0 * N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / (-t$95$5)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.00000000000000011e-206

   1. Initial program 51.1%

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

   4. Applied rewrites 71.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift--.f64 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

      20. sqrt-prod N/A

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

   6. Applied rewrites 86.2%

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

   if -4.00000000000000011e-206 < (/.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))) < 2e216

   1. Initial program 22.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.2

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

   5. Applied rewrites 34.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 33.6%

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

   if 2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.6%

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0 \cdot \frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 36.7

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + 0 \cdot \color{blue}{\frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 36.7%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(2 + 0 \cdot \frac{A}{C}\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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 A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 16.4

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

   5. Applied rewrites 16.4%

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

   7. Applied rewrites 23.5%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := -t\_2\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_3}\\ t_5 := \sqrt{\left(2 \cdot F\right) \cdot t\_0}\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-206}:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{C \cdot 2}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- t_0))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3 (- t_2))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_3))
        (t_5 (sqrt (* (* 2.0 F) t_0))))
   (if (<= t_4 -4e-206)
     (* t_5 (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) t_1))
     (if (<= t_4 2e+216)
       (/
        (sqrt
         (*
          (* (fma (* C A) -4.0 (* B_m B_m)) 2.0)
          (* F (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)))))
        t_3)
       (if (<= t_4 INFINITY)
         (* t_5 (/ (sqrt (* C 2.0)) t_1))
         (*
          (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot C B_m) C)))
          (- (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = -t_2;
	double t_4 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_3;
	double t_5 = sqrt(((2.0 * F) * t_0));
	double tmp;
	if (t_4 <= -4e-206) {
		tmp = t_5 * (sqrt(((hypot((A - C), B_m) + A) + C)) / t_1);
	} else if (t_4 <= 2e+216) {
		tmp = sqrt(((fma((C * A), -4.0, (B_m * B_m)) * 2.0) * (F * fma(-0.5, ((B_m * B_m) / A), (C * 2.0))))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_5 * (sqrt((C * 2.0)) / t_1);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(C, B_m) + C))) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(-t_2)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_3)
	t_5 = sqrt(Float64(Float64(2.0 * F) * t_0))
	tmp = 0.0
	if (t_4 <= -4e-206)
		tmp = Float64(t_5 * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / t_1));
	elseif (t_4 <= 2e+216)
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0) * Float64(F * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0))))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(t_5 * Float64(sqrt(Float64(C * 2.0)) / t_1));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(C, B_m) + C))) * Float64(-sqrt(F)));
	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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-t$95$2)}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -4e-206], N[(t$95$5 * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+216], N[(N[Sqrt[N[(N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$5 * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.00000000000000011e-206

   1. Initial program 51.1%

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

   4. Applied rewrites 71.9%

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

   if -4.00000000000000011e-206 < (/.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))) < 2e216

   1. Initial program 22.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.2

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

   5. Applied rewrites 34.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 33.6%

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

   if 2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.6%

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0 \cdot \frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 36.7

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + 0 \cdot \color{blue}{\frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 36.7%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(2 + 0 \cdot \frac{A}{C}\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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 A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 16.4

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

   5. Applied rewrites 16.4%

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

   7. Applied rewrites 23.5%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_2 := -t\_0\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ t_5 := \sqrt{\left(2 \cdot F\right) \cdot t\_0}\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-206}:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right) \cdot \left(\left(t\_1 \cdot F\right) \cdot 2\right)}}{-t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_5 \cdot \frac{\sqrt{C \cdot 2}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{\mathsf{hypot}\left(C, B\_m\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m)))
        (t_2 (- t_0))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3)))
        (t_5 (sqrt (* (* 2.0 F) t_0))))
   (if (<= t_4 -4e-206)
     (* t_5 (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) t_2))
     (if (<= t_4 2e+216)
       (/
        (sqrt (* (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)) (* (* t_1 F) 2.0)))
        (- t_1))
       (if (<= t_4 INFINITY)
         (* t_5 (/ (sqrt (* C 2.0)) t_2))
         (*
          (* (/ (sqrt 2.0) B_m) (sqrt (+ (hypot C B_m) C)))
          (- (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double t_2 = -t_0;
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_3;
	double t_5 = sqrt(((2.0 * F) * t_0));
	double tmp;
	if (t_4 <= -4e-206) {
		tmp = t_5 * (sqrt(((hypot((A - C), B_m) + A) + C)) / t_2);
	} else if (t_4 <= 2e+216) {
		tmp = sqrt((fma(-0.5, ((B_m * B_m) / A), (C * 2.0)) * ((t_1 * F) * 2.0))) / -t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_5 * (sqrt((C * 2.0)) / t_2);
	} else {
		tmp = ((sqrt(2.0) / B_m) * sqrt((hypot(C, B_m) + C))) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_2 = Float64(-t_0)
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_3))
	t_5 = sqrt(Float64(Float64(2.0 * F) * t_0))
	tmp = 0.0
	if (t_4 <= -4e-206)
		tmp = Float64(t_5 * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / t_2));
	elseif (t_4 <= 2e+216)
		tmp = Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)) * Float64(Float64(t_1 * F) * 2.0))) / Float64(-t_1));
	elseif (t_4 <= Inf)
		tmp = Float64(t_5 * Float64(sqrt(Float64(C * 2.0)) / t_2));
	else
		tmp = Float64(Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(hypot(C, B_m) + C))) * Float64(-sqrt(F)));
	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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -4e-206], N[(t$95$5 * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+216], N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * F), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$5 * N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.00000000000000011e-206

   1. Initial program 51.1%

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

   4. Applied rewrites 71.9%

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

   if -4.00000000000000011e-206 < (/.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))) < 2e216

   1. Initial program 22.5%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 34.2

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

   5. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 34.2%

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

   if 2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.6%

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + \color{blue}{0 \cdot \frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 36.7

         \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{C \cdot \left(2 + 0 \cdot \color{blue}{\frac{A}{C}}\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 36.7%

      \[\leadsto \left(-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \cdot \frac{\sqrt{\color{blue}{C \cdot \left(2 + 0 \cdot \frac{A}{C}\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   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 A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 16.4

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

   5. Applied rewrites 16.4%

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

   7. Applied rewrites 23.5%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -4 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 2 \cdot 10^{+216}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{C \cdot 2}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\mathsf{hypot}\left(C, B\right) + C}\right) \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-180}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

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
 (if (<= (pow B_m 2.0) 1e-180)
   (/ (sqrt (* -16.0 (* (* (* C C) F) A))) (- (fma (* -4.0 C) A (* B_m B_m))))
   (* (* (sqrt (+ B_m C)) (- (sqrt F))) (/ (sqrt 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 tmp;
	if (pow(B_m, 2.0) <= 1e-180) {
		tmp = sqrt((-16.0 * (((C * C) * F) * A))) / -fma((-4.0 * C), A, (B_m * B_m));
	} else {
		tmp = (sqrt((B_m + C)) * -sqrt(F)) * (sqrt(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)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-180)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(Float64(C * C) * F) * A))) / Float64(-fma(Float64(-4.0 * C), A, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))) * Float64(sqrt(2.0) / 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[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-180], N[(N[Sqrt[N[(-16.0 * N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e-180

   1. Initial program 20.7%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 12.9%

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

   if 1e-180 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 20.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 22.5

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

   5. Applied rewrites 22.5%

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

   7. Applied rewrites 29.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 23.9%

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

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-180}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 720:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(2 \cdot C\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

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 720.0)
     (/ (sqrt (* (* t_0 (* F 2.0)) (* 2.0 C))) (- t_0))
     (* (* (sqrt (+ (hypot C B_m) C)) (sqrt F)) (/ (sqrt 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 <= 720.0) {
		tmp = sqrt(((t_0 * (F * 2.0)) * (2.0 * C))) / -t_0;
	} else {
		tmp = (sqrt((hypot(C, B_m) + C)) * sqrt(F)) * (sqrt(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 <= 720.0)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * Float64(2.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(hypot(C, B_m) + C)) * sqrt(F)) * Float64(sqrt(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, 720.0], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 720

   1. Initial program 23.2%

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

   4. Applied rewrites 35.7%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.2

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 3.4%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 14.2

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

   12. Applied rewrites 14.2%

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

   if 720 < B

   1. Initial program 11.6%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 41.2

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

   5. Applied rewrites 41.2%

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

   7. Applied rewrites 57.0%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 720:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(2 \cdot C\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 720:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(2 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 3.1 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(C, B\_m\right) + C} \cdot \frac{\sqrt{F \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

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 720.0)
     (/ (sqrt (* (* t_0 (* F 2.0)) (* 2.0 C))) (- t_0))
     (if (<= B_m 3.1e+207)
       (* (sqrt (+ (hypot C B_m) C)) (/ (sqrt (* F 2.0)) (- B_m)))
       (* (* (sqrt (+ B_m C)) (- (sqrt F))) (/ (sqrt 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 <= 720.0) {
		tmp = sqrt(((t_0 * (F * 2.0)) * (2.0 * C))) / -t_0;
	} else if (B_m <= 3.1e+207) {
		tmp = sqrt((hypot(C, B_m) + C)) * (sqrt((F * 2.0)) / -B_m);
	} else {
		tmp = (sqrt((B_m + C)) * -sqrt(F)) * (sqrt(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 <= 720.0)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * Float64(2.0 * C))) / Float64(-t_0));
	elseif (B_m <= 3.1e+207)
		tmp = Float64(sqrt(Float64(hypot(C, B_m) + C)) * Float64(sqrt(Float64(F * 2.0)) / Float64(-B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))) * Float64(sqrt(2.0) / 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, 720.0], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 3.1e+207], N[(N[Sqrt[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 720

   1. Initial program 23.2%

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

   4. Applied rewrites 35.7%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.2

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 3.4%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 14.2

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

   12. Applied rewrites 14.2%

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

   if 720 < B < 3.1000000000000003e207

   1. Initial program 16.1%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 35.2

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

   5. Applied rewrites 35.2%

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

   7. Applied rewrites 42.7%

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

   9. Applied rewrites 43.2%

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

   if 3.1000000000000003e207 < B

   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 A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 56.7

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

   5. Applied rewrites 56.7%

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

   7. Applied rewrites 94.3%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 94.3%

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

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 720:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(2 \cdot C\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \frac{\sqrt{F \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 720:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(2 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 2.35 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

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 720.0)
     (/ (sqrt (* (* t_0 (* F 2.0)) (* 2.0 C))) (- t_0))
     (if (<= B_m 2.35e+151)
       (/ (sqrt (* 2.0 (* (+ (hypot C B_m) C) F))) (- B_m))
       (* (* (sqrt (+ B_m C)) (- (sqrt F))) (/ (sqrt 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 <= 720.0) {
		tmp = sqrt(((t_0 * (F * 2.0)) * (2.0 * C))) / -t_0;
	} else if (B_m <= 2.35e+151) {
		tmp = sqrt((2.0 * ((hypot(C, B_m) + C) * F))) / -B_m;
	} else {
		tmp = (sqrt((B_m + C)) * -sqrt(F)) * (sqrt(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 <= 720.0)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * Float64(2.0 * C))) / Float64(-t_0));
	elseif (B_m <= 2.35e+151)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(hypot(C, B_m) + C) * F))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))) * Float64(sqrt(2.0) / 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, 720.0], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 2.35e+151], N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 720

   1. Initial program 23.2%

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

   4. Applied rewrites 35.7%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.2

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 3.4%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 14.2

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

   12. Applied rewrites 14.2%

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

   if 720 < B < 2.34999999999999995e151

   1. Initial program 22.1%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 40.8

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

   5. Applied rewrites 40.8%

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

   7. Applied rewrites 40.8%

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

   if 2.34999999999999995e151 < B

   1. Initial program 0.1%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 41.6

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

   5. Applied rewrites 41.6%

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

   7. Applied rewrites 70.1%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 69.6%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 720:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(2 \cdot C\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{+151}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 720:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(2 \cdot C\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B\_m + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

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 720.0)
     (/ (sqrt (* (* t_0 (* F 2.0)) (* 2.0 C))) (- t_0))
     (* (* (sqrt (+ B_m C)) (- (sqrt F))) (/ (sqrt 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 <= 720.0) {
		tmp = sqrt(((t_0 * (F * 2.0)) * (2.0 * C))) / -t_0;
	} else {
		tmp = (sqrt((B_m + C)) * -sqrt(F)) * (sqrt(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 <= 720.0)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * Float64(2.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))) * Float64(sqrt(2.0) / 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, 720.0], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 720

   1. Initial program 23.2%

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

   4. Applied rewrites 35.7%

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

   5. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.2

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 3.4%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 14.2

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

   12. Applied rewrites 14.2%

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

   if 720 < B

   1. Initial program 11.6%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 41.2

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

   5. Applied rewrites 41.2%

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

   7. Applied rewrites 57.0%

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

   8. Taylor expanded in C around 0

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

   10. Applied rewrites 51.5%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 720:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(2 \cdot C\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right) \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.5% accurate, 8.6× speedup

Math

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

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 (+ B_m C)) (- (sqrt F))) (/ (sqrt 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((B_m + C)) * -sqrt(F)) * (sqrt(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((b_m + c)) * -sqrt(f)) * (sqrt(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((B_m + C)) * -Math.sqrt(F)) * (Math.sqrt(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((B_m + C)) * -math.sqrt(F)) * (math.sqrt(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 Float64(Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))) * Float64(sqrt(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((B_m + C)) * -sqrt(F)) * (sqrt(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[(N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in A around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. unpow2 N/A

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

   13. lower-hypot.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sqrt.f64 16.3

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

5. Applied rewrites 16.3%

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

7. Applied rewrites 20.3%

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

8. Taylor expanded in C around 0

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

10. Applied rewrites 16.1%

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

11. Final simplification 16.1%

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

Alternative 13: 27.2% accurate, 12.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 3.05 \cdot 10^{-235}:\\ \;\;\;\;-\frac{2}{B\_m} \cdot \sqrt{C \cdot F}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 3.05e-235)
   (- (* (/ 2.0 B_m) (sqrt (* C F))))
   (- (sqrt (* (/ F B_m) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 3.05e-235) {
		tmp = -((2.0 / B_m) * sqrt((C * F)));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 3.05d-235) then
        tmp = -((2.0d0 / b_m) * sqrt((c * f)))
    else
        tmp = -sqrt(((f / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 3.05e-235) {
		tmp = -((2.0 / B_m) * Math.sqrt((C * F)));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 3.05e-235:
		tmp = -((2.0 / B_m) * math.sqrt((C * F)))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 3.05e-235)
		tmp = Float64(-Float64(Float64(2.0 / B_m) * sqrt(Float64(C * F))));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 3.05e-235)
		tmp = -((2.0 / B_m) * sqrt((C * F)));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 3.05e-235], (-N[(N[(2.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 3.04999999999999994e-235

   1. Initial program 26.5%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 9.0

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

   5. Applied rewrites 9.0%

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

   7. Applied rewrites 8.6%

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

   9. Applied rewrites 7.5%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 5.1%

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

   if 3.04999999999999994e-235 < F

   1. Initial program 18.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 16.1

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

   5. Applied rewrites 16.1%

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

   7. Applied rewrites 16.1%

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

Alternative 14: 35.9% accurate, 12.6× speedup

Math

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

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)) (- (sqrt B_m))))

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 12.9

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

5. Applied rewrites 12.9%

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

7. Applied rewrites 16.3%

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

8. Final simplification 16.3%

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

Alternative 15: 35.9% accurate, 12.6× speedup

Math

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

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)) (sqrt (/ 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) * sqrt((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) * sqrt((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) * Math.sqrt((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) * math.sqrt((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 Float64(Float64(-sqrt(F)) * sqrt(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) * sqrt((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[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 12.9

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

5. Applied rewrites 12.9%

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

7. Applied rewrites 12.9%

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

9. Applied rewrites 16.3%

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

10. Final simplification 16.3%

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

Alternative 16: 27.1% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.3%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 12.9

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

5. Applied rewrites 12.9%

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

7. Applied rewrites 12.9%

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

Alternative 17: 27.1% accurate, 16.9× 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 Float64(-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 20.3%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 12.9

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

5. Applied rewrites 12.9%

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

7. Applied rewrites 12.9%

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

9. Applied rewrites 12.9%

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

Reproduce

herbie shell --seed 2024351 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))