ABCF->ab-angle a

Percentage Accurate: 18.7% → 64.4%

Time: 10.0s

Alternatives: 16

Speedup: 9.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.7% accurate, 1.0× speedup

Math

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

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: 64.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|B\right| \cdot \left|B\right|\\ t_1 := \mathsf{max}\left(A, C\right) - \mathsf{min}\left(A, C\right)\\ t_2 := {\left(\left|B\right|\right)}^{2}\\ t_3 := t\_2 - \left(4 \cdot \mathsf{min}\left(A, C\right)\right) \cdot \mathsf{max}\left(A, C\right)\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(\mathsf{min}\left(A, C\right) + \mathsf{max}\left(A, C\right)\right) + \sqrt{{\left(\mathsf{min}\left(A, C\right) - \mathsf{max}\left(A, C\right)\right)}^{2} + t\_2}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{min}\left(A, C\right)}{\sqrt{-16 \cdot \mathsf{min}\left(A, C\right)}} \cdot \frac{4}{\sqrt{F}}}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-212}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(-8, \mathsf{max}\left(A, C\right) \cdot \mathsf{min}\left(A, C\right), t\_0 \cdot 2\right)} \cdot \sqrt{\left(\sqrt{\mathsf{fma}\left(t\_1, t\_1, t\_0\right)} + \left(\mathsf{max}\left(A, C\right) + \mathsf{min}\left(A, C\right)\right)\right) \cdot F}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{\mathsf{max}\left(A, C\right)}{\left|\mathsf{max}\left(A, C\right)\right| \cdot \sqrt{\frac{F}{\mathsf{min}\left(A, C\right)} \cdot -16}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (fabs B) (fabs B)))
        (t_1 (- (fmax A C) (fmin A C)))
        (t_2 (pow (fabs B) 2.0))
        (t_3 (- t_2 (* (* 4.0 (fmin A C)) (fmax A C))))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+
              (+ (fmin A C) (fmax A C))
              (sqrt (+ (pow (- (fmin A C) (fmax A C)) 2.0) t_2))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ 1.0 (* (/ (fmin A C) (sqrt (* -16.0 (fmin A C)))) (/ 4.0 (sqrt F))))
     (if (<= t_4 -5e-212)
       (/
        (-
         (*
          (sqrt (fma -8.0 (* (fmax A C) (fmin A C)) (* t_0 2.0)))
          (sqrt (* (+ (sqrt (fma t_1 t_1 t_0)) (+ (fmax A C) (fmin A C))) F))))
        t_3)
       (if (<= t_4 0.0)
         (*
          -0.25
          (/
           1.0
           (/
            (fmax A C)
            (* (fabs (fmax A C)) (sqrt (* (/ F (fmin A C)) -16.0))))))
         (if (<= t_4 INFINITY)
           (* 0.25 (* (sqrt (* -16.0 F)) (sqrt (/ 1.0 (fmin A C)))))
           (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B)))))))))))

C

double code(double A, double B, double C, double F) {
	double t_0 = fabs(B) * fabs(B);
	double t_1 = fmax(A, C) - fmin(A, C);
	double t_2 = pow(fabs(B), 2.0);
	double t_3 = t_2 - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((fmin(A, C) + fmax(A, C)) + sqrt((pow((fmin(A, C) - fmax(A, C)), 2.0) + t_2))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = 1.0 / ((fmin(A, C) / sqrt((-16.0 * fmin(A, C)))) * (4.0 / sqrt(F)));
	} else if (t_4 <= -5e-212) {
		tmp = -(sqrt(fma(-8.0, (fmax(A, C) * fmin(A, C)), (t_0 * 2.0))) * sqrt(((sqrt(fma(t_1, t_1, t_0)) + (fmax(A, C) + fmin(A, C))) * F))) / t_3;
	} else if (t_4 <= 0.0) {
		tmp = -0.25 * (1.0 / (fmax(A, C) / (fabs(fmax(A, C)) * sqrt(((F / fmin(A, C)) * -16.0)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = 0.25 * (sqrt((-16.0 * F)) * sqrt((1.0 / fmin(A, C))));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	t_0 = Float64(abs(B) * abs(B))
	t_1 = Float64(fmax(A, C) - fmin(A, C))
	t_2 = abs(B) ^ 2.0
	t_3 = Float64(t_2 - Float64(Float64(4.0 * fmin(A, C)) * fmax(A, C)))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(fmin(A, C) + fmax(A, C)) + sqrt(Float64((Float64(fmin(A, C) - fmax(A, C)) ^ 2.0) + t_2)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(Float64(fmin(A, C) / sqrt(Float64(-16.0 * fmin(A, C)))) * Float64(4.0 / sqrt(F))));
	elseif (t_4 <= -5e-212)
		tmp = Float64(Float64(-Float64(sqrt(fma(-8.0, Float64(fmax(A, C) * fmin(A, C)), Float64(t_0 * 2.0))) * sqrt(Float64(Float64(sqrt(fma(t_1, t_1, t_0)) + Float64(fmax(A, C) + fmin(A, C))) * F)))) / t_3);
	elseif (t_4 <= 0.0)
		tmp = Float64(-0.25 * Float64(1.0 / Float64(fmax(A, C) / Float64(abs(fmax(A, C)) * sqrt(Float64(Float64(F / fmin(A, C)) * -16.0))))));
	elseif (t_4 <= Inf)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * F)) * sqrt(Float64(1.0 / fmin(A, C)))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Abs[B], $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[A, C], $MachinePrecision] - N[Min[A, C], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[(N[(4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Min[A, C], $MachinePrecision] + N[Max[A, C], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[Min[A, C], $MachinePrecision] - N[Max[A, C], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(1.0 / N[(N[(N[Min[A, C], $MachinePrecision] / N[Sqrt[N[(-16.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-212], N[((-N[(N[Sqrt[N[(-8.0 * N[(N[Max[A, C], $MachinePrecision] * N[Min[A, C], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[N[(t$95$1 * t$95$1 + t$95$0), $MachinePrecision]], $MachinePrecision] + N[(N[Max[A, C], $MachinePrecision] + N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(-0.25 * N[(1.0 / N[(N[Max[A, C], $MachinePrecision] / N[(N[Abs[N[Max[A, C], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(0.25 * N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 18.7%

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

   3. Applied rewrites 18.9%

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

   4. Taylor expanded in C around inf

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      5. lower-*.f64 19.3%

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

   6. Applied rewrites 19.3%

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{\frac{4 \cdot A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\color{blue}{-16 \cdot \left(A \cdot F\right)}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\left(-16 \cdot A\right) \cdot F}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\left(-16 \cdot A\right) \cdot F}}} \]

      10. sqrt-unprod N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \color{blue}{\sqrt{F}}}} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \sqrt{\color{blue}{F}}}} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \sqrt{F}}} \]

      13. times-frac N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \frac{\color{blue}{4}}{\sqrt{F}}} \]

      16. lower-/.f64 18.0%

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \frac{4}{\color{blue}{\sqrt{F}}}} \]

   8. Applied rewrites 18.0%

      \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

   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))) < -5.00000000000000043e-212

   1. Initial program 18.7%

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

   3. Applied rewrites 20.5%

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

   if -5.00000000000000043e-212 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      7. lower-pow.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      2. div-flip N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      4. lower-unsound-/.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\color{blue}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{C \cdot C} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      17. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      19. lower-fabs.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      20. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      22. lower-/.f64 15.4%

          \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

   6. Applied rewrites 15.4%

      \[\leadsto -0.25 \cdot \frac{1}{\color{blue}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}}} \]

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

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot A\right)}}{A} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot A}}{A} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      7. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      8. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      10. lower-unsound-sqrt.f64 5.9%

          \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   6. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   7. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      5. lower-/.f64 5.9%

         \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

   9. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]

   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 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{B}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 64.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|B\right| \cdot \left|B\right|\\ t_1 := \mathsf{max}\left(A, C\right) - \mathsf{min}\left(A, C\right)\\ t_2 := {\left(\left|B\right|\right)}^{2}\\ t_3 := t\_2 - \left(4 \cdot \mathsf{min}\left(A, C\right)\right) \cdot \mathsf{max}\left(A, C\right)\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(\mathsf{min}\left(A, C\right) + \mathsf{max}\left(A, C\right)\right) + \sqrt{{\left(\mathsf{min}\left(A, C\right) - \mathsf{max}\left(A, C\right)\right)}^{2} + t\_2}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{min}\left(A, C\right)}{\sqrt{-16 \cdot \mathsf{min}\left(A, C\right)}} \cdot \frac{4}{\sqrt{F}}}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4 \cdot \mathsf{min}\left(A, C\right), \mathsf{max}\left(A, C\right), t\_0\right) \cdot F} \cdot \frac{\sqrt{\left(\sqrt{\mathsf{fma}\left(t\_1, t\_1, t\_0\right)} + \left(\mathsf{max}\left(A, C\right) + \mathsf{min}\left(A, C\right)\right)\right) \cdot 2}}{\left(\mathsf{min}\left(A, C\right) \cdot 4\right) \cdot \mathsf{max}\left(A, C\right) - t\_0}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{\mathsf{max}\left(A, C\right)}{\left|\mathsf{max}\left(A, C\right)\right| \cdot \sqrt{\frac{F}{\mathsf{min}\left(A, C\right)} \cdot -16}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (fabs B) (fabs B)))
        (t_1 (- (fmax A C) (fmin A C)))
        (t_2 (pow (fabs B) 2.0))
        (t_3 (- t_2 (* (* 4.0 (fmin A C)) (fmax A C))))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+
              (+ (fmin A C) (fmax A C))
              (sqrt (+ (pow (- (fmin A C) (fmax A C)) 2.0) t_2))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ 1.0 (* (/ (fmin A C) (sqrt (* -16.0 (fmin A C)))) (/ 4.0 (sqrt F))))
     (if (<= t_4 -5e-212)
       (*
        (sqrt (* (fma (* -4.0 (fmin A C)) (fmax A C) t_0) F))
        (/
         (sqrt (* (+ (sqrt (fma t_1 t_1 t_0)) (+ (fmax A C) (fmin A C))) 2.0))
         (- (* (* (fmin A C) 4.0) (fmax A C)) t_0)))
       (if (<= t_4 0.0)
         (*
          -0.25
          (/
           1.0
           (/
            (fmax A C)
            (* (fabs (fmax A C)) (sqrt (* (/ F (fmin A C)) -16.0))))))
         (if (<= t_4 INFINITY)
           (* 0.25 (* (sqrt (* -16.0 F)) (sqrt (/ 1.0 (fmin A C)))))
           (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B)))))))))))

C

double code(double A, double B, double C, double F) {
	double t_0 = fabs(B) * fabs(B);
	double t_1 = fmax(A, C) - fmin(A, C);
	double t_2 = pow(fabs(B), 2.0);
	double t_3 = t_2 - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((fmin(A, C) + fmax(A, C)) + sqrt((pow((fmin(A, C) - fmax(A, C)), 2.0) + t_2))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = 1.0 / ((fmin(A, C) / sqrt((-16.0 * fmin(A, C)))) * (4.0 / sqrt(F)));
	} else if (t_4 <= -5e-212) {
		tmp = sqrt((fma((-4.0 * fmin(A, C)), fmax(A, C), t_0) * F)) * (sqrt(((sqrt(fma(t_1, t_1, t_0)) + (fmax(A, C) + fmin(A, C))) * 2.0)) / (((fmin(A, C) * 4.0) * fmax(A, C)) - t_0));
	} else if (t_4 <= 0.0) {
		tmp = -0.25 * (1.0 / (fmax(A, C) / (fabs(fmax(A, C)) * sqrt(((F / fmin(A, C)) * -16.0)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = 0.25 * (sqrt((-16.0 * F)) * sqrt((1.0 / fmin(A, C))));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	t_0 = Float64(abs(B) * abs(B))
	t_1 = Float64(fmax(A, C) - fmin(A, C))
	t_2 = abs(B) ^ 2.0
	t_3 = Float64(t_2 - Float64(Float64(4.0 * fmin(A, C)) * fmax(A, C)))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(fmin(A, C) + fmax(A, C)) + sqrt(Float64((Float64(fmin(A, C) - fmax(A, C)) ^ 2.0) + t_2)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(Float64(fmin(A, C) / sqrt(Float64(-16.0 * fmin(A, C)))) * Float64(4.0 / sqrt(F))));
	elseif (t_4 <= -5e-212)
		tmp = Float64(sqrt(Float64(fma(Float64(-4.0 * fmin(A, C)), fmax(A, C), t_0) * F)) * Float64(sqrt(Float64(Float64(sqrt(fma(t_1, t_1, t_0)) + Float64(fmax(A, C) + fmin(A, C))) * 2.0)) / Float64(Float64(Float64(fmin(A, C) * 4.0) * fmax(A, C)) - t_0)));
	elseif (t_4 <= 0.0)
		tmp = Float64(-0.25 * Float64(1.0 / Float64(fmax(A, C) / Float64(abs(fmax(A, C)) * sqrt(Float64(Float64(F / fmin(A, C)) * -16.0))))));
	elseif (t_4 <= Inf)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * F)) * sqrt(Float64(1.0 / fmin(A, C)))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Abs[B], $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Max[A, C], $MachinePrecision] - N[Min[A, C], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[(N[(4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Min[A, C], $MachinePrecision] + N[Max[A, C], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[Min[A, C], $MachinePrecision] - N[Max[A, C], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(1.0 / N[(N[(N[Min[A, C], $MachinePrecision] / N[Sqrt[N[(-16.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-212], N[(N[Sqrt[N[(N[(N[(-4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision] + t$95$0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[N[(t$95$1 * t$95$1 + t$95$0), $MachinePrecision]], $MachinePrecision] + N[(N[Max[A, C], $MachinePrecision] + N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Min[A, C], $MachinePrecision] * 4.0), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(-0.25 * N[(1.0 / N[(N[Max[A, C], $MachinePrecision] / N[(N[Abs[N[Max[A, C], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(0.25 * N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 18.7%

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

   3. Applied rewrites 18.9%

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

   4. Taylor expanded in C around inf

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      5. lower-*.f64 19.3%

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

   6. Applied rewrites 19.3%

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{\frac{4 \cdot A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\color{blue}{-16 \cdot \left(A \cdot F\right)}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\left(-16 \cdot A\right) \cdot F}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\left(-16 \cdot A\right) \cdot F}}} \]

      10. sqrt-unprod N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \color{blue}{\sqrt{F}}}} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \sqrt{\color{blue}{F}}}} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \sqrt{F}}} \]

      13. times-frac N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \frac{\color{blue}{4}}{\sqrt{F}}} \]

      16. lower-/.f64 18.0%

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \frac{4}{\color{blue}{\sqrt{F}}}} \]

   8. Applied rewrites 18.0%

      \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

   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))) < -5.00000000000000043e-212

   1. Initial program 18.7%

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

   3. Applied rewrites 18.9%

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

   5. Applied rewrites 21.8%

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

   if -5.00000000000000043e-212 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      7. lower-pow.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      2. div-flip N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      4. lower-unsound-/.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\color{blue}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{C \cdot C} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      17. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      19. lower-fabs.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      20. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      22. lower-/.f64 15.4%

          \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

   6. Applied rewrites 15.4%

      \[\leadsto -0.25 \cdot \frac{1}{\color{blue}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}}} \]

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

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot A\right)}}{A} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot A}}{A} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      7. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      8. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      10. lower-unsound-sqrt.f64 5.9%

          \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   6. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   7. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      5. lower-/.f64 5.9%

         \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

   9. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]

   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 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{B}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 64.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(A, C\right) - \mathsf{min}\left(A, C\right)\\ t_1 := \left|B\right| \cdot \left|B\right|\\ t_2 := {\left(\left|B\right|\right)}^{2}\\ t_3 := t\_2 - \left(4 \cdot \mathsf{min}\left(A, C\right)\right) \cdot \mathsf{max}\left(A, C\right)\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(\mathsf{min}\left(A, C\right) + \mathsf{max}\left(A, C\right)\right) + \sqrt{{\left(\mathsf{min}\left(A, C\right) - \mathsf{max}\left(A, C\right)\right)}^{2} + t\_2}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{min}\left(A, C\right)}{\sqrt{-16 \cdot \mathsf{min}\left(A, C\right)}} \cdot \frac{4}{\sqrt{F}}}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-212}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\sqrt{\mathsf{fma}\left(t\_0, t\_0, t\_1\right)} + \left(\mathsf{max}\left(A, C\right) + \mathsf{min}\left(A, C\right)\right)\right) \cdot \left(F + F\right)\right) \cdot \mathsf{fma}\left(\mathsf{max}\left(A, C\right) \cdot -4, \mathsf{min}\left(A, C\right), t\_1\right)}}{\mathsf{max}\left(A, C\right) \cdot \left(\mathsf{min}\left(A, C\right) \cdot 4\right) - t\_1}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{\mathsf{max}\left(A, C\right)}{\left|\mathsf{max}\left(A, C\right)\right| \cdot \sqrt{\frac{F}{\mathsf{min}\left(A, C\right)} \cdot -16}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (fmax A C) (fmin A C)))
        (t_1 (* (fabs B) (fabs B)))
        (t_2 (pow (fabs B) 2.0))
        (t_3 (- t_2 (* (* 4.0 (fmin A C)) (fmax A C))))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+
              (+ (fmin A C) (fmax A C))
              (sqrt (+ (pow (- (fmin A C) (fmax A C)) 2.0) t_2))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ 1.0 (* (/ (fmin A C) (sqrt (* -16.0 (fmin A C)))) (/ 4.0 (sqrt F))))
     (if (<= t_4 -5e-212)
       (/
        (sqrt
         (*
          (* (+ (sqrt (fma t_0 t_0 t_1)) (+ (fmax A C) (fmin A C))) (+ F F))
          (fma (* (fmax A C) -4.0) (fmin A C) t_1)))
        (- (* (fmax A C) (* (fmin A C) 4.0)) t_1))
       (if (<= t_4 0.0)
         (*
          -0.25
          (/
           1.0
           (/
            (fmax A C)
            (* (fabs (fmax A C)) (sqrt (* (/ F (fmin A C)) -16.0))))))
         (if (<= t_4 INFINITY)
           (* 0.25 (* (sqrt (* -16.0 F)) (sqrt (/ 1.0 (fmin A C)))))
           (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B)))))))))))

C

double code(double A, double B, double C, double F) {
	double t_0 = fmax(A, C) - fmin(A, C);
	double t_1 = fabs(B) * fabs(B);
	double t_2 = pow(fabs(B), 2.0);
	double t_3 = t_2 - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((fmin(A, C) + fmax(A, C)) + sqrt((pow((fmin(A, C) - fmax(A, C)), 2.0) + t_2))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = 1.0 / ((fmin(A, C) / sqrt((-16.0 * fmin(A, C)))) * (4.0 / sqrt(F)));
	} else if (t_4 <= -5e-212) {
		tmp = sqrt((((sqrt(fma(t_0, t_0, t_1)) + (fmax(A, C) + fmin(A, C))) * (F + F)) * fma((fmax(A, C) * -4.0), fmin(A, C), t_1))) / ((fmax(A, C) * (fmin(A, C) * 4.0)) - t_1);
	} else if (t_4 <= 0.0) {
		tmp = -0.25 * (1.0 / (fmax(A, C) / (fabs(fmax(A, C)) * sqrt(((F / fmin(A, C)) * -16.0)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = 0.25 * (sqrt((-16.0 * F)) * sqrt((1.0 / fmin(A, C))));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	t_0 = Float64(fmax(A, C) - fmin(A, C))
	t_1 = Float64(abs(B) * abs(B))
	t_2 = abs(B) ^ 2.0
	t_3 = Float64(t_2 - Float64(Float64(4.0 * fmin(A, C)) * fmax(A, C)))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(fmin(A, C) + fmax(A, C)) + sqrt(Float64((Float64(fmin(A, C) - fmax(A, C)) ^ 2.0) + t_2)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(Float64(fmin(A, C) / sqrt(Float64(-16.0 * fmin(A, C)))) * Float64(4.0 / sqrt(F))));
	elseif (t_4 <= -5e-212)
		tmp = Float64(sqrt(Float64(Float64(Float64(sqrt(fma(t_0, t_0, t_1)) + Float64(fmax(A, C) + fmin(A, C))) * Float64(F + F)) * fma(Float64(fmax(A, C) * -4.0), fmin(A, C), t_1))) / Float64(Float64(fmax(A, C) * Float64(fmin(A, C) * 4.0)) - t_1));
	elseif (t_4 <= 0.0)
		tmp = Float64(-0.25 * Float64(1.0 / Float64(fmax(A, C) / Float64(abs(fmax(A, C)) * sqrt(Float64(Float64(F / fmin(A, C)) * -16.0))))));
	elseif (t_4 <= Inf)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * F)) * sqrt(Float64(1.0 / fmin(A, C)))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Max[A, C], $MachinePrecision] - N[Min[A, C], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[B], $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[(N[(4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Min[A, C], $MachinePrecision] + N[Max[A, C], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[Min[A, C], $MachinePrecision] - N[Max[A, C], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(1.0 / N[(N[(N[Min[A, C], $MachinePrecision] / N[Sqrt[N[(-16.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-212], N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(t$95$0 * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] + N[(N[Max[A, C], $MachinePrecision] + N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F + F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Max[A, C], $MachinePrecision] * -4.0), $MachinePrecision] * N[Min[A, C], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[Max[A, C], $MachinePrecision] * N[(N[Min[A, C], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(-0.25 * N[(1.0 / N[(N[Max[A, C], $MachinePrecision] / N[(N[Abs[N[Max[A, C], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(0.25 * N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 18.7%

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

   3. Applied rewrites 18.9%

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

   4. Taylor expanded in C around inf

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      5. lower-*.f64 19.3%

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

   6. Applied rewrites 19.3%

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{\frac{4 \cdot A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\color{blue}{-16 \cdot \left(A \cdot F\right)}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\left(-16 \cdot A\right) \cdot F}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\left(-16 \cdot A\right) \cdot F}}} \]

      10. sqrt-unprod N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \color{blue}{\sqrt{F}}}} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \sqrt{\color{blue}{F}}}} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \sqrt{F}}} \]

      13. times-frac N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \frac{\color{blue}{4}}{\sqrt{F}}} \]

      16. lower-/.f64 18.0%

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \frac{4}{\color{blue}{\sqrt{F}}}} \]

   8. Applied rewrites 18.0%

      \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

   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))) < -5.00000000000000043e-212

   1. Initial program 18.7%

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

   3. Applied rewrites 18.9%

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

   if -5.00000000000000043e-212 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      7. lower-pow.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      2. div-flip N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      4. lower-unsound-/.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\color{blue}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{C \cdot C} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      17. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      19. lower-fabs.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      20. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      22. lower-/.f64 15.4%

          \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

   6. Applied rewrites 15.4%

      \[\leadsto -0.25 \cdot \frac{1}{\color{blue}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}}} \]

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

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot A\right)}}{A} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot A}}{A} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      7. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      8. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      10. lower-unsound-sqrt.f64 5.9%

          \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   6. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   7. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      5. lower-/.f64 5.9%

         \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

   9. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]

   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 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{B}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|B\right| \cdot \left|B\right|\\ t_1 := {\left(\left|B\right|\right)}^{2}\\ t_2 := t\_1 - \left(4 \cdot \mathsf{min}\left(A, C\right)\right) \cdot \mathsf{max}\left(A, C\right)\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(\mathsf{min}\left(A, C\right) + \mathsf{max}\left(A, C\right)\right) + \sqrt{{\left(\mathsf{min}\left(A, C\right) - \mathsf{max}\left(A, C\right)\right)}^{2} + t\_1}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{min}\left(A, C\right)}{\sqrt{-16 \cdot \mathsf{min}\left(A, C\right)}} \cdot \frac{4}{\sqrt{F}}}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{max}\left(A, C\right) \cdot \left(\mathsf{min}\left(A, C\right) \cdot 4\right) - t\_0}{\sqrt{\left(\left(\sqrt{\mathsf{fma}\left(\mathsf{max}\left(A, C\right), \mathsf{max}\left(A, C\right), t\_0\right)} + \left(\mathsf{max}\left(A, C\right) + \mathsf{min}\left(A, C\right)\right)\right) \cdot \left(F + F\right)\right) \cdot \mathsf{fma}\left(\mathsf{max}\left(A, C\right) \cdot -4, \mathsf{min}\left(A, C\right), t\_0\right)}}}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{\mathsf{max}\left(A, C\right)}{\left|\mathsf{max}\left(A, C\right)\right| \cdot \sqrt{\frac{F}{\mathsf{min}\left(A, C\right)} \cdot -16}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (fabs B) (fabs B)))
        (t_1 (pow (fabs B) 2.0))
        (t_2 (- t_1 (* (* 4.0 (fmin A C)) (fmax A C))))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+
              (+ (fmin A C) (fmax A C))
              (sqrt (+ (pow (- (fmin A C) (fmax A C)) 2.0) t_1))))))
          t_2)))
   (if (<= t_3 (- INFINITY))
     (/ 1.0 (* (/ (fmin A C) (sqrt (* -16.0 (fmin A C)))) (/ 4.0 (sqrt F))))
     (if (<= t_3 -5e-212)
       (/
        1.0
        (/
         (- (* (fmax A C) (* (fmin A C) 4.0)) t_0)
         (sqrt
          (*
           (*
            (+
             (sqrt (fma (fmax A C) (fmax A C) t_0))
             (+ (fmax A C) (fmin A C)))
            (+ F F))
           (fma (* (fmax A C) -4.0) (fmin A C) t_0)))))
       (if (<= t_3 0.0)
         (*
          -0.25
          (/
           1.0
           (/
            (fmax A C)
            (* (fabs (fmax A C)) (sqrt (* (/ F (fmin A C)) -16.0))))))
         (if (<= t_3 INFINITY)
           (* 0.25 (* (sqrt (* -16.0 F)) (sqrt (/ 1.0 (fmin A C)))))
           (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B)))))))))))

C

double code(double A, double B, double C, double F) {
	double t_0 = fabs(B) * fabs(B);
	double t_1 = pow(fabs(B), 2.0);
	double t_2 = t_1 - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((fmin(A, C) + fmax(A, C)) + sqrt((pow((fmin(A, C) - fmax(A, C)), 2.0) + t_1))))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = 1.0 / ((fmin(A, C) / sqrt((-16.0 * fmin(A, C)))) * (4.0 / sqrt(F)));
	} else if (t_3 <= -5e-212) {
		tmp = 1.0 / (((fmax(A, C) * (fmin(A, C) * 4.0)) - t_0) / sqrt((((sqrt(fma(fmax(A, C), fmax(A, C), t_0)) + (fmax(A, C) + fmin(A, C))) * (F + F)) * fma((fmax(A, C) * -4.0), fmin(A, C), t_0))));
	} else if (t_3 <= 0.0) {
		tmp = -0.25 * (1.0 / (fmax(A, C) / (fabs(fmax(A, C)) * sqrt(((F / fmin(A, C)) * -16.0)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = 0.25 * (sqrt((-16.0 * F)) * sqrt((1.0 / fmin(A, C))));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	t_0 = Float64(abs(B) * abs(B))
	t_1 = abs(B) ^ 2.0
	t_2 = Float64(t_1 - Float64(Float64(4.0 * fmin(A, C)) * fmax(A, C)))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(fmin(A, C) + fmax(A, C)) + sqrt(Float64((Float64(fmin(A, C) - fmax(A, C)) ^ 2.0) + t_1)))))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(Float64(fmin(A, C) / sqrt(Float64(-16.0 * fmin(A, C)))) * Float64(4.0 / sqrt(F))));
	elseif (t_3 <= -5e-212)
		tmp = Float64(1.0 / Float64(Float64(Float64(fmax(A, C) * Float64(fmin(A, C) * 4.0)) - t_0) / sqrt(Float64(Float64(Float64(sqrt(fma(fmax(A, C), fmax(A, C), t_0)) + Float64(fmax(A, C) + fmin(A, C))) * Float64(F + F)) * fma(Float64(fmax(A, C) * -4.0), fmin(A, C), t_0)))));
	elseif (t_3 <= 0.0)
		tmp = Float64(-0.25 * Float64(1.0 / Float64(fmax(A, C) / Float64(abs(fmax(A, C)) * sqrt(Float64(Float64(F / fmin(A, C)) * -16.0))))));
	elseif (t_3 <= Inf)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * F)) * sqrt(Float64(1.0 / fmin(A, C)))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Abs[B], $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[(N[(4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Min[A, C], $MachinePrecision] + N[Max[A, C], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[Min[A, C], $MachinePrecision] - N[Max[A, C], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(1.0 / N[(N[(N[Min[A, C], $MachinePrecision] / N[Sqrt[N[(-16.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(4.0 / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-212], N[(1.0 / N[(N[(N[(N[Max[A, C], $MachinePrecision] * N[(N[Min[A, C], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Sqrt[N[(N[Max[A, C], $MachinePrecision] * N[Max[A, C], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] + N[(N[Max[A, C], $MachinePrecision] + N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F + F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Max[A, C], $MachinePrecision] * -4.0), $MachinePrecision] * N[Min[A, C], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(-0.25 * N[(1.0 / N[(N[Max[A, C], $MachinePrecision] / N[(N[Abs[N[Max[A, C], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(0.25 * N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 18.7%

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

   3. Applied rewrites 18.9%

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

   4. Taylor expanded in C around inf

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      5. lower-*.f64 19.3%

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

   6. Applied rewrites 19.3%

      \[\leadsto \frac{1}{\color{blue}{4 \cdot \frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \color{blue}{\frac{A}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{4 \cdot \frac{A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{1}{\frac{4 \cdot A}{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\color{blue}{-16 \cdot \left(A \cdot F\right)}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot \left(A \cdot F\right)}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\left(-16 \cdot A\right) \cdot F}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{\left(-16 \cdot A\right) \cdot F}}} \]

      10. sqrt-unprod N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \color{blue}{\sqrt{F}}}} \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \sqrt{\color{blue}{F}}}} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{A \cdot 4}{\sqrt{-16 \cdot A} \cdot \sqrt{F}}} \]

      13. times-frac N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \frac{\color{blue}{4}}{\sqrt{F}}} \]

      16. lower-/.f64 18.0%

          \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \frac{4}{\color{blue}{\sqrt{F}}}} \]

   8. Applied rewrites 18.0%

      \[\leadsto \frac{1}{\frac{A}{\sqrt{-16 \cdot A}} \cdot \color{blue}{\frac{4}{\sqrt{F}}}} \]

   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))) < -5.00000000000000043e-212

   1. Initial program 18.7%

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

   3. Applied rewrites 18.9%

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

   4. Taylor expanded in A around 0

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

   6. Applied rewrites 14.3%

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

   7. Taylor expanded in A around 0

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

   9. Applied rewrites 15.1%

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

   if -5.00000000000000043e-212 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      7. lower-pow.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      2. div-flip N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      4. lower-unsound-/.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\color{blue}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{C \cdot C} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      17. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      19. lower-fabs.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      20. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      22. lower-/.f64 15.4%

          \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

   6. Applied rewrites 15.4%

      \[\leadsto -0.25 \cdot \frac{1}{\color{blue}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}}} \]

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

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot A\right)}}{A} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot A}}{A} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      7. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      8. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      10. lower-unsound-sqrt.f64 5.9%

          \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   6. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   7. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      5. lower-/.f64 5.9%

         \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

   9. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]

   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 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{B}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 59.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|B\right| \cdot \left|B\right|\\ t_1 := \mathsf{min}\left(A, C\right) + \mathsf{max}\left(A, C\right)\\ t_2 := {\left(\left|B\right|\right)}^{2}\\ t_3 := t\_2 - \left(4 \cdot \mathsf{min}\left(A, C\right)\right) \cdot \mathsf{max}\left(A, C\right)\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(t\_1 + \sqrt{{\left(\mathsf{min}\left(A, C\right) - \mathsf{max}\left(A, C\right)\right)}^{2} + t\_2}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -6 \cdot 10^{+130}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-212}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{max}\left(A, C\right) \cdot \left(\mathsf{min}\left(A, C\right) \cdot 4\right) - t\_0}{\sqrt{\mathsf{fma}\left(-4 \cdot \mathsf{min}\left(A, C\right), \mathsf{max}\left(A, C\right), t\_0\right)} \cdot \sqrt{\left(\mathsf{fma}\left(t\_1, \frac{F}{\left|B\right|}, F\right) \cdot 2\right) \cdot \left|B\right|}}}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{\mathsf{max}\left(A, C\right)}{\left|\mathsf{max}\left(A, C\right)\right| \cdot \sqrt{\frac{F}{\mathsf{min}\left(A, C\right)} \cdot -16}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (fabs B) (fabs B)))
        (t_1 (+ (fmin A C) (fmax A C)))
        (t_2 (pow (fabs B) 2.0))
        (t_3 (- t_2 (* (* 4.0 (fmin A C)) (fmax A C))))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ t_1 (sqrt (+ (pow (- (fmin A C) (fmax A C)) 2.0) t_2))))))
          t_3)))
   (if (<= t_4 -6e+130)
     (* -0.25 (* (sqrt F) (sqrt (/ -16.0 (fmin A C)))))
     (if (<= t_4 -5e-212)
       (/
        1.0
        (/
         (- (* (fmax A C) (* (fmin A C) 4.0)) t_0)
         (*
          (sqrt (fma (* -4.0 (fmin A C)) (fmax A C) t_0))
          (sqrt (* (* (fma t_1 (/ F (fabs B)) F) 2.0) (fabs B))))))
       (if (<= t_4 0.0)
         (*
          -0.25
          (/
           1.0
           (/
            (fmax A C)
            (* (fabs (fmax A C)) (sqrt (* (/ F (fmin A C)) -16.0))))))
         (if (<= t_4 INFINITY)
           (* 0.25 (* (sqrt (* -16.0 F)) (sqrt (/ 1.0 (fmin A C)))))
           (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B)))))))))))

C

double code(double A, double B, double C, double F) {
	double t_0 = fabs(B) * fabs(B);
	double t_1 = fmin(A, C) + fmax(A, C);
	double t_2 = pow(fabs(B), 2.0);
	double t_3 = t_2 - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * (t_1 + sqrt((pow((fmin(A, C) - fmax(A, C)), 2.0) + t_2))))) / t_3;
	double tmp;
	if (t_4 <= -6e+130) {
		tmp = -0.25 * (sqrt(F) * sqrt((-16.0 / fmin(A, C))));
	} else if (t_4 <= -5e-212) {
		tmp = 1.0 / (((fmax(A, C) * (fmin(A, C) * 4.0)) - t_0) / (sqrt(fma((-4.0 * fmin(A, C)), fmax(A, C), t_0)) * sqrt(((fma(t_1, (F / fabs(B)), F) * 2.0) * fabs(B)))));
	} else if (t_4 <= 0.0) {
		tmp = -0.25 * (1.0 / (fmax(A, C) / (fabs(fmax(A, C)) * sqrt(((F / fmin(A, C)) * -16.0)))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = 0.25 * (sqrt((-16.0 * F)) * sqrt((1.0 / fmin(A, C))));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	t_0 = Float64(abs(B) * abs(B))
	t_1 = Float64(fmin(A, C) + fmax(A, C))
	t_2 = abs(B) ^ 2.0
	t_3 = Float64(t_2 - Float64(Float64(4.0 * fmin(A, C)) * fmax(A, C)))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(t_1 + sqrt(Float64((Float64(fmin(A, C) - fmax(A, C)) ^ 2.0) + t_2)))))) / t_3)
	tmp = 0.0
	if (t_4 <= -6e+130)
		tmp = Float64(-0.25 * Float64(sqrt(F) * sqrt(Float64(-16.0 / fmin(A, C)))));
	elseif (t_4 <= -5e-212)
		tmp = Float64(1.0 / Float64(Float64(Float64(fmax(A, C) * Float64(fmin(A, C) * 4.0)) - t_0) / Float64(sqrt(fma(Float64(-4.0 * fmin(A, C)), fmax(A, C), t_0)) * sqrt(Float64(Float64(fma(t_1, Float64(F / abs(B)), F) * 2.0) * abs(B))))));
	elseif (t_4 <= 0.0)
		tmp = Float64(-0.25 * Float64(1.0 / Float64(fmax(A, C) / Float64(abs(fmax(A, C)) * sqrt(Float64(Float64(F / fmin(A, C)) * -16.0))))));
	elseif (t_4 <= Inf)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * F)) * sqrt(Float64(1.0 / fmin(A, C)))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Abs[B], $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Min[A, C], $MachinePrecision] + N[Max[A, C], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[(N[(4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[Power[N[(N[Min[A, C], $MachinePrecision] - N[Max[A, C], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -6e+130], N[(-0.25 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(-16.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-212], N[(1.0 / N[(N[(N[(N[Max[A, C], $MachinePrecision] * N[(N[Min[A, C], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(t$95$1 * N[(F / N[Abs[B], $MachinePrecision]), $MachinePrecision] + F), $MachinePrecision] * 2.0), $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(-0.25 * N[(1.0 / N[(N[Max[A, C], $MachinePrecision] / N[(N[Abs[N[Max[A, C], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(0.25 * N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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))) < -5.9999999999999999e130

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot A\right) \cdot F}}{A} \]

      5. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      6. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      7. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      9. lower-unsound-sqrt.f64 18.0%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   6. Applied rewrites 18.0%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{-16}{A}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      5. lower-/.f64 18.0%

         \[\leadsto -0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

   9. Applied rewrites 18.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]

   if -5.9999999999999999e130 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000043e-212

   1. Initial program 18.7%

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

   3. Applied rewrites 18.9%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 7.3%

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

   6. Applied rewrites 7.3%

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

   8. Applied rewrites 8.1%

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

   if -5.00000000000000043e-212 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      7. lower-pow.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      2. div-flip N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      4. lower-unsound-/.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\color{blue}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{C \cdot C} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      17. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      19. lower-fabs.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      20. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      22. lower-/.f64 15.4%

          \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

   6. Applied rewrites 15.4%

      \[\leadsto -0.25 \cdot \frac{1}{\color{blue}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}}} \]

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

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot A\right)}}{A} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot A}}{A} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      7. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      8. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      10. lower-unsound-sqrt.f64 5.9%

          \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   6. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   7. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      5. lower-/.f64 5.9%

         \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

   9. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]

   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 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{B}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 59.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(A, C\right) + \mathsf{max}\left(A, C\right)\\ t_1 := {\left(\left|B\right|\right)}^{2}\\ t_2 := t\_1 - \left(4 \cdot \mathsf{min}\left(A, C\right)\right) \cdot \mathsf{max}\left(A, C\right)\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(t\_0 + \sqrt{{\left(\mathsf{min}\left(A, C\right) - \mathsf{max}\left(A, C\right)\right)}^{2} + t\_1}\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(-4 \cdot \mathsf{min}\left(A, C\right), \mathsf{max}\left(A, C\right), \left|B\right| \cdot \left|B\right|\right)\\ \mathbf{if}\;t\_3 \leq -6 \cdot 10^{+130}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-212}:\\ \;\;\;\;\frac{-1}{t\_4} \cdot \sqrt{\left(\left(\mathsf{fma}\left(t\_0, \frac{F}{\left|B\right|}, F\right) \cdot 2\right) \cdot \left|B\right|\right) \cdot t\_4}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{\mathsf{max}\left(A, C\right)}{\left|\mathsf{max}\left(A, C\right)\right| \cdot \sqrt{\frac{F}{\mathsf{min}\left(A, C\right)} \cdot -16}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (fmin A C) (fmax A C)))
        (t_1 (pow (fabs B) 2.0))
        (t_2 (- t_1 (* (* 4.0 (fmin A C)) (fmax A C))))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ t_0 (sqrt (+ (pow (- (fmin A C) (fmax A C)) 2.0) t_1))))))
          t_2))
        (t_4 (fma (* -4.0 (fmin A C)) (fmax A C) (* (fabs B) (fabs B)))))
   (if (<= t_3 -6e+130)
     (* -0.25 (* (sqrt F) (sqrt (/ -16.0 (fmin A C)))))
     (if (<= t_3 -5e-212)
       (*
        (/ -1.0 t_4)
        (sqrt (* (* (* (fma t_0 (/ F (fabs B)) F) 2.0) (fabs B)) t_4)))
       (if (<= t_3 0.0)
         (*
          -0.25
          (/
           1.0
           (/
            (fmax A C)
            (* (fabs (fmax A C)) (sqrt (* (/ F (fmin A C)) -16.0))))))
         (if (<= t_3 INFINITY)
           (* 0.25 (* (sqrt (* -16.0 F)) (sqrt (/ 1.0 (fmin A C)))))
           (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B)))))))))))

C

double code(double A, double B, double C, double F) {
	double t_0 = fmin(A, C) + fmax(A, C);
	double t_1 = pow(fabs(B), 2.0);
	double t_2 = t_1 - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * (t_0 + sqrt((pow((fmin(A, C) - fmax(A, C)), 2.0) + t_1))))) / t_2;
	double t_4 = fma((-4.0 * fmin(A, C)), fmax(A, C), (fabs(B) * fabs(B)));
	double tmp;
	if (t_3 <= -6e+130) {
		tmp = -0.25 * (sqrt(F) * sqrt((-16.0 / fmin(A, C))));
	} else if (t_3 <= -5e-212) {
		tmp = (-1.0 / t_4) * sqrt((((fma(t_0, (F / fabs(B)), F) * 2.0) * fabs(B)) * t_4));
	} else if (t_3 <= 0.0) {
		tmp = -0.25 * (1.0 / (fmax(A, C) / (fabs(fmax(A, C)) * sqrt(((F / fmin(A, C)) * -16.0)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = 0.25 * (sqrt((-16.0 * F)) * sqrt((1.0 / fmin(A, C))));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	t_0 = Float64(fmin(A, C) + fmax(A, C))
	t_1 = abs(B) ^ 2.0
	t_2 = Float64(t_1 - Float64(Float64(4.0 * fmin(A, C)) * fmax(A, C)))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(t_0 + sqrt(Float64((Float64(fmin(A, C) - fmax(A, C)) ^ 2.0) + t_1)))))) / t_2)
	t_4 = fma(Float64(-4.0 * fmin(A, C)), fmax(A, C), Float64(abs(B) * abs(B)))
	tmp = 0.0
	if (t_3 <= -6e+130)
		tmp = Float64(-0.25 * Float64(sqrt(F) * sqrt(Float64(-16.0 / fmin(A, C)))));
	elseif (t_3 <= -5e-212)
		tmp = Float64(Float64(-1.0 / t_4) * sqrt(Float64(Float64(Float64(fma(t_0, Float64(F / abs(B)), F) * 2.0) * abs(B)) * t_4)));
	elseif (t_3 <= 0.0)
		tmp = Float64(-0.25 * Float64(1.0 / Float64(fmax(A, C) / Float64(abs(fmax(A, C)) * sqrt(Float64(Float64(F / fmin(A, C)) * -16.0))))));
	elseif (t_3 <= Inf)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * F)) * sqrt(Float64(1.0 / fmin(A, C)))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Min[A, C], $MachinePrecision] + N[Max[A, C], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[(N[(4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[Power[N[(N[Min[A, C], $MachinePrecision] - N[Max[A, C], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision] + N[(N[Abs[B], $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -6e+130], N[(-0.25 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(-16.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-212], N[(N[(-1.0 / t$95$4), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(t$95$0 * N[(F / N[Abs[B], $MachinePrecision]), $MachinePrecision] + F), $MachinePrecision] * 2.0), $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(-0.25 * N[(1.0 / N[(N[Max[A, C], $MachinePrecision] / N[(N[Abs[N[Max[A, C], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(0.25 * N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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))) < -5.9999999999999999e130

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot A\right) \cdot F}}{A} \]

      5. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      6. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      7. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      9. lower-unsound-sqrt.f64 18.0%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   6. Applied rewrites 18.0%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{-16}{A}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      5. lower-/.f64 18.0%

         \[\leadsto -0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

   9. Applied rewrites 18.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]

   if -5.9999999999999999e130 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000043e-212

   1. Initial program 18.7%

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

   3. Applied rewrites 18.9%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 7.3%

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

   6. Applied rewrites 7.3%

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

   8. Applied rewrites 7.1%

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

   if -5.00000000000000043e-212 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      7. lower-pow.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      2. div-flip N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      4. lower-unsound-/.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\color{blue}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{C \cdot C} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      17. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      19. lower-fabs.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      20. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      22. lower-/.f64 15.4%

          \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

   6. Applied rewrites 15.4%

      \[\leadsto -0.25 \cdot \frac{1}{\color{blue}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}}} \]

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

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot A\right)}}{A} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot A}}{A} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      7. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      8. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      10. lower-unsound-sqrt.f64 5.9%

          \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   6. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   7. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      5. lower-/.f64 5.9%

         \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

   9. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]

   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 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{B}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 59.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(4 \cdot \mathsf{min}\left(A, C\right)\right) \cdot \mathsf{max}\left(A, C\right)\\ t_1 := \left|B\right| \cdot \left|B\right|\\ t_2 := \mathsf{min}\left(A, C\right) + \mathsf{max}\left(A, C\right)\\ t_3 := {\left(\left|B\right|\right)}^{2}\\ t_4 := t\_3 - t\_0\\ t_5 := \frac{-\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(t\_2 + \sqrt{{\left(\mathsf{min}\left(A, C\right) - \mathsf{max}\left(A, C\right)\right)}^{2} + t\_3}\right)}}{t\_4}\\ \mathbf{if}\;t\_5 \leq -6 \cdot 10^{+130}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{elif}\;t\_5 \leq -5 \cdot 10^{-212}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(t\_2, \frac{F}{\left|B\right|}, F\right) \cdot 2\right) \cdot \left|B\right|\right) \cdot \mathsf{fma}\left(-4 \cdot \mathsf{min}\left(A, C\right), \mathsf{max}\left(A, C\right), t\_1\right)}}{t\_0 - t\_1}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{\mathsf{max}\left(A, C\right)}{\left|\mathsf{max}\left(A, C\right)\right| \cdot \sqrt{\frac{F}{\mathsf{min}\left(A, C\right)} \cdot -16}}}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 (fmin A C)) (fmax A C)))
        (t_1 (* (fabs B) (fabs B)))
        (t_2 (+ (fmin A C) (fmax A C)))
        (t_3 (pow (fabs B) 2.0))
        (t_4 (- t_3 t_0))
        (t_5
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_4 F))
             (+ t_2 (sqrt (+ (pow (- (fmin A C) (fmax A C)) 2.0) t_3))))))
          t_4)))
   (if (<= t_5 -6e+130)
     (* -0.25 (* (sqrt F) (sqrt (/ -16.0 (fmin A C)))))
     (if (<= t_5 -5e-212)
       (/
        (sqrt
         (*
          (* (* (fma t_2 (/ F (fabs B)) F) 2.0) (fabs B))
          (fma (* -4.0 (fmin A C)) (fmax A C) t_1)))
        (- t_0 t_1))
       (if (<= t_5 0.0)
         (*
          -0.25
          (/
           1.0
           (/
            (fmax A C)
            (* (fabs (fmax A C)) (sqrt (* (/ F (fmin A C)) -16.0))))))
         (if (<= t_5 INFINITY)
           (* 0.25 (* (sqrt (* -16.0 F)) (sqrt (/ 1.0 (fmin A C)))))
           (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B)))))))))))

C

double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * fmin(A, C)) * fmax(A, C);
	double t_1 = fabs(B) * fabs(B);
	double t_2 = fmin(A, C) + fmax(A, C);
	double t_3 = pow(fabs(B), 2.0);
	double t_4 = t_3 - t_0;
	double t_5 = -sqrt(((2.0 * (t_4 * F)) * (t_2 + sqrt((pow((fmin(A, C) - fmax(A, C)), 2.0) + t_3))))) / t_4;
	double tmp;
	if (t_5 <= -6e+130) {
		tmp = -0.25 * (sqrt(F) * sqrt((-16.0 / fmin(A, C))));
	} else if (t_5 <= -5e-212) {
		tmp = sqrt((((fma(t_2, (F / fabs(B)), F) * 2.0) * fabs(B)) * fma((-4.0 * fmin(A, C)), fmax(A, C), t_1))) / (t_0 - t_1);
	} else if (t_5 <= 0.0) {
		tmp = -0.25 * (1.0 / (fmax(A, C) / (fabs(fmax(A, C)) * sqrt(((F / fmin(A, C)) * -16.0)))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = 0.25 * (sqrt((-16.0 * F)) * sqrt((1.0 / fmin(A, C))));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * fmin(A, C)) * fmax(A, C))
	t_1 = Float64(abs(B) * abs(B))
	t_2 = Float64(fmin(A, C) + fmax(A, C))
	t_3 = abs(B) ^ 2.0
	t_4 = Float64(t_3 - t_0)
	t_5 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(t_2 + sqrt(Float64((Float64(fmin(A, C) - fmax(A, C)) ^ 2.0) + t_3)))))) / t_4)
	tmp = 0.0
	if (t_5 <= -6e+130)
		tmp = Float64(-0.25 * Float64(sqrt(F) * sqrt(Float64(-16.0 / fmin(A, C)))));
	elseif (t_5 <= -5e-212)
		tmp = Float64(sqrt(Float64(Float64(Float64(fma(t_2, Float64(F / abs(B)), F) * 2.0) * abs(B)) * fma(Float64(-4.0 * fmin(A, C)), fmax(A, C), t_1))) / Float64(t_0 - t_1));
	elseif (t_5 <= 0.0)
		tmp = Float64(-0.25 * Float64(1.0 / Float64(fmax(A, C) / Float64(abs(fmax(A, C)) * sqrt(Float64(Float64(F / fmin(A, C)) * -16.0))))));
	elseif (t_5 <= Inf)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * F)) * sqrt(Float64(1.0 / fmin(A, C)))));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[B], $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Min[A, C], $MachinePrecision] + N[Max[A, C], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 + N[Sqrt[N[(N[Power[N[(N[Min[A, C], $MachinePrecision] - N[Max[A, C], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, -6e+130], N[(-0.25 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(-16.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -5e-212], N[(N[Sqrt[N[(N[(N[(N[(t$95$2 * N[(F / N[Abs[B], $MachinePrecision]), $MachinePrecision] + F), $MachinePrecision] * 2.0), $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(-0.25 * N[(1.0 / N[(N[Max[A, C], $MachinePrecision] / N[(N[Abs[N[Max[A, C], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(0.25 * N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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))) < -5.9999999999999999e130

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot A\right) \cdot F}}{A} \]

      5. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      6. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      7. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      9. lower-unsound-sqrt.f64 18.0%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   6. Applied rewrites 18.0%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{-16}{A}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      5. lower-/.f64 18.0%

         \[\leadsto -0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

   9. Applied rewrites 18.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]

   if -5.9999999999999999e130 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000043e-212

   1. Initial program 18.7%

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

   3. Applied rewrites 18.9%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 7.3%

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

   6. Applied rewrites 7.3%

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

   8. Applied rewrites 7.1%

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

   if -5.00000000000000043e-212 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      7. lower-pow.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      2. div-flip N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      4. lower-unsound-/.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\color{blue}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{C \cdot C} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      17. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      19. lower-fabs.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      20. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      22. lower-/.f64 15.4%

          \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

   6. Applied rewrites 15.4%

      \[\leadsto -0.25 \cdot \frac{1}{\color{blue}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}}} \]

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

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot A\right)}}{A} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot A}}{A} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      7. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      8. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

      10. lower-unsound-sqrt.f64 5.9%

          \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   6. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{A}}{A} \]

   7. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

      5. lower-/.f64 5.9%

         \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \sqrt{\frac{1}{A}}\right) \]

   9. Applied rewrites 5.9%

      \[\leadsto 0.25 \cdot \left(\sqrt{-16 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{A}}}\right) \]

   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 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{B}}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 56.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := -0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{\mathsf{min}\left(A, C\right)}}\right)\\ t_1 := -1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ t_2 := {\left(\left|B\right|\right)}^{2}\\ t_3 := t\_2 - \left(4 \cdot \mathsf{min}\left(A, C\right)\right) \cdot \mathsf{max}\left(A, C\right)\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(\mathsf{min}\left(A, C\right) + \mathsf{max}\left(A, C\right)\right) + \sqrt{{\left(\mathsf{min}\left(A, C\right) - \mathsf{max}\left(A, C\right)\right)}^{2} + t\_2}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -6 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{\mathsf{min}\left(A, C\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* -0.25 (* (sqrt F) (sqrt (/ -16.0 (fmin A C))))))
        (t_1 (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B))))))
        (t_2 (pow (fabs B) 2.0))
        (t_3 (- t_2 (* (* 4.0 (fmin A C)) (fmax A C))))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+
              (+ (fmin A C) (fmax A C))
              (sqrt (+ (pow (- (fmin A C) (fmax A C)) 2.0) t_2))))))
          t_3)))
   (if (<= t_4 -6e+130)
     t_0
     (if (<= t_4 -5e-212)
       t_1
       (if (<= t_4 0.0)
         t_0
         (if (<= t_4 INFINITY)
           (* 0.25 (sqrt (* -16.0 (/ F (fmin A C)))))
           t_1))))))

C

double code(double A, double B, double C, double F) {
	double t_0 = -0.25 * (sqrt(F) * sqrt((-16.0 / fmin(A, C))));
	double t_1 = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	double t_2 = pow(fabs(B), 2.0);
	double t_3 = t_2 - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((fmin(A, C) + fmax(A, C)) + sqrt((pow((fmin(A, C) - fmax(A, C)), 2.0) + t_2))))) / t_3;
	double tmp;
	if (t_4 <= -6e+130) {
		tmp = t_0;
	} else if (t_4 <= -5e-212) {
		tmp = t_1;
	} else if (t_4 <= 0.0) {
		tmp = t_0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = 0.25 * sqrt((-16.0 * (F / fmin(A, C))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = -0.25 * (Math.sqrt(F) * Math.sqrt((-16.0 / fmin(A, C))));
	double t_1 = -1.0 * (Math.sqrt(F) * Math.sqrt((2.0 / Math.abs(B))));
	double t_2 = Math.pow(Math.abs(B), 2.0);
	double t_3 = t_2 - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_4 = -Math.sqrt(((2.0 * (t_3 * F)) * ((fmin(A, C) + fmax(A, C)) + Math.sqrt((Math.pow((fmin(A, C) - fmax(A, C)), 2.0) + t_2))))) / t_3;
	double tmp;
	if (t_4 <= -6e+130) {
		tmp = t_0;
	} else if (t_4 <= -5e-212) {
		tmp = t_1;
	} else if (t_4 <= 0.0) {
		tmp = t_0;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = 0.25 * Math.sqrt((-16.0 * (F / fmin(A, C))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(A, B, C, F):
	t_0 = -0.25 * (math.sqrt(F) * math.sqrt((-16.0 / fmin(A, C))))
	t_1 = -1.0 * (math.sqrt(F) * math.sqrt((2.0 / math.fabs(B))))
	t_2 = math.pow(math.fabs(B), 2.0)
	t_3 = t_2 - ((4.0 * fmin(A, C)) * fmax(A, C))
	t_4 = -math.sqrt(((2.0 * (t_3 * F)) * ((fmin(A, C) + fmax(A, C)) + math.sqrt((math.pow((fmin(A, C) - fmax(A, C)), 2.0) + t_2))))) / t_3
	tmp = 0
	if t_4 <= -6e+130:
		tmp = t_0
	elif t_4 <= -5e-212:
		tmp = t_1
	elif t_4 <= 0.0:
		tmp = t_0
	elif t_4 <= math.inf:
		tmp = 0.25 * math.sqrt((-16.0 * (F / fmin(A, C))))
	else:
		tmp = t_1
	return tmp

Julia

function code(A, B, C, F)
	t_0 = Float64(-0.25 * Float64(sqrt(F) * sqrt(Float64(-16.0 / fmin(A, C)))))
	t_1 = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))))
	t_2 = abs(B) ^ 2.0
	t_3 = Float64(t_2 - Float64(Float64(4.0 * fmin(A, C)) * fmax(A, C)))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(fmin(A, C) + fmax(A, C)) + sqrt(Float64((Float64(fmin(A, C) - fmax(A, C)) ^ 2.0) + t_2)))))) / t_3)
	tmp = 0.0
	if (t_4 <= -6e+130)
		tmp = t_0;
	elseif (t_4 <= -5e-212)
		tmp = t_1;
	elseif (t_4 <= 0.0)
		tmp = t_0;
	elseif (t_4 <= Inf)
		tmp = Float64(0.25 * sqrt(Float64(-16.0 * Float64(F / fmin(A, C)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(A, B, C, F)
	t_0 = -0.25 * (sqrt(F) * sqrt((-16.0 / min(A, C))));
	t_1 = -1.0 * (sqrt(F) * sqrt((2.0 / abs(B))));
	t_2 = abs(B) ^ 2.0;
	t_3 = t_2 - ((4.0 * min(A, C)) * max(A, C));
	t_4 = -sqrt(((2.0 * (t_3 * F)) * ((min(A, C) + max(A, C)) + sqrt((((min(A, C) - max(A, C)) ^ 2.0) + t_2))))) / t_3;
	tmp = 0.0;
	if (t_4 <= -6e+130)
		tmp = t_0;
	elseif (t_4 <= -5e-212)
		tmp = t_1;
	elseif (t_4 <= 0.0)
		tmp = t_0;
	elseif (t_4 <= Inf)
		tmp = 0.25 * sqrt((-16.0 * (F / min(A, C))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-0.25 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(-16.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[(N[(4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Min[A, C], $MachinePrecision] + N[Max[A, C], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[Min[A, C], $MachinePrecision] - N[Max[A, C], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -6e+130], t$95$0, If[LessEqual[t$95$4, -5e-212], t$95$1, If[LessEqual[t$95$4, 0.0], t$95$0, If[LessEqual[t$95$4, Infinity], N[(0.25 * N[Sqrt[N[(-16.0 * N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.9999999999999999e130 or -5.00000000000000043e-212 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot A\right) \cdot F}}{A} \]

      5. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      6. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      7. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      9. lower-unsound-sqrt.f64 18.0%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   6. Applied rewrites 18.0%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{-16}{A}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      5. lower-/.f64 18.0%

         \[\leadsto -0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

   9. Applied rewrites 18.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]

   if -5.9999999999999999e130 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000043e-212 or +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 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

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

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

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

   5. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      3. lower-/.f64 10.4%

         \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

   7. Applied rewrites 10.4%

      \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|B\right| \leq 3.15 \cdot 10^{-279}:\\ \;\;\;\;-0.25 \cdot \frac{1}{\frac{\mathsf{max}\left(A, C\right)}{\left|\mathsf{max}\left(A, C\right)\right| \cdot \sqrt{\frac{F}{\mathsf{min}\left(A, C\right)} \cdot -16}}}\\ \mathbf{elif}\;\left|B\right| \leq 430:\\ \;\;\;\;0.25 \cdot \frac{\sqrt{-16 \cdot \left(\mathsf{min}\left(A, C\right) \cdot F\right)}}{\mathsf{min}\left(A, C\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (if (<= (fabs B) 3.15e-279)
   (*
    -0.25
    (/
     1.0
     (/ (fmax A C) (* (fabs (fmax A C)) (sqrt (* (/ F (fmin A C)) -16.0))))))
   (if (<= (fabs B) 430.0)
     (* 0.25 (/ (sqrt (* -16.0 (* (fmin A C) F))) (fmin A C)))
     (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B))))))))

C

double code(double A, double B, double C, double F) {
	double tmp;
	if (fabs(B) <= 3.15e-279) {
		tmp = -0.25 * (1.0 / (fmax(A, C) / (fabs(fmax(A, C)) * sqrt(((F / fmin(A, C)) * -16.0)))));
	} else if (fabs(B) <= 430.0) {
		tmp = 0.25 * (sqrt((-16.0 * (fmin(A, C) * F))) / fmin(A, C));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

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) :: tmp
    if (abs(b) <= 3.15d-279) then
        tmp = (-0.25d0) * (1.0d0 / (fmax(a, c) / (abs(fmax(a, c)) * sqrt(((f / fmin(a, c)) * (-16.0d0))))))
    else if (abs(b) <= 430.0d0) then
        tmp = 0.25d0 * (sqrt(((-16.0d0) * (fmin(a, c) * f))) / fmin(a, c))
    else
        tmp = (-1.0d0) * (sqrt(f) * sqrt((2.0d0 / abs(b))))
    end if
    code = tmp
end function

Java

public static double code(double A, double B, double C, double F) {
	double tmp;
	if (Math.abs(B) <= 3.15e-279) {
		tmp = -0.25 * (1.0 / (fmax(A, C) / (Math.abs(fmax(A, C)) * Math.sqrt(((F / fmin(A, C)) * -16.0)))));
	} else if (Math.abs(B) <= 430.0) {
		tmp = 0.25 * (Math.sqrt((-16.0 * (fmin(A, C) * F))) / fmin(A, C));
	} else {
		tmp = -1.0 * (Math.sqrt(F) * Math.sqrt((2.0 / Math.abs(B))));
	}
	return tmp;
}

Python

def code(A, B, C, F):
	tmp = 0
	if math.fabs(B) <= 3.15e-279:
		tmp = -0.25 * (1.0 / (fmax(A, C) / (math.fabs(fmax(A, C)) * math.sqrt(((F / fmin(A, C)) * -16.0)))))
	elif math.fabs(B) <= 430.0:
		tmp = 0.25 * (math.sqrt((-16.0 * (fmin(A, C) * F))) / fmin(A, C))
	else:
		tmp = -1.0 * (math.sqrt(F) * math.sqrt((2.0 / math.fabs(B))))
	return tmp

Julia

function code(A, B, C, F)
	tmp = 0.0
	if (abs(B) <= 3.15e-279)
		tmp = Float64(-0.25 * Float64(1.0 / Float64(fmax(A, C) / Float64(abs(fmax(A, C)) * sqrt(Float64(Float64(F / fmin(A, C)) * -16.0))))));
	elseif (abs(B) <= 430.0)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * Float64(fmin(A, C) * F))) / fmin(A, C)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (abs(B) <= 3.15e-279)
		tmp = -0.25 * (1.0 / (max(A, C) / (abs(max(A, C)) * sqrt(((F / min(A, C)) * -16.0)))));
	elseif (abs(B) <= 430.0)
		tmp = 0.25 * (sqrt((-16.0 * (min(A, C) * F))) / min(A, C));
	else
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / abs(B))));
	end
	tmp_2 = tmp;
end

Wolfram

code[A_, B_, C_, F_] := If[LessEqual[N[Abs[B], $MachinePrecision], 3.15e-279], N[(-0.25 * N[(1.0 / N[(N[Max[A, C], $MachinePrecision] / N[(N[Abs[N[Max[A, C], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[B], $MachinePrecision], 430.0], N[(0.25 * N[(N[Sqrt[N[(-16.0 * N[(N[Min[A, C], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.1499999999999999e-279

   1. Initial program 18.7%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      7. lower-pow.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      2. div-flip N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\color{blue}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      4. lower-unsound-/.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\color{blue}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}} \]

      7. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{\frac{{C}^{2} \cdot F}{A} \cdot -16}}} \]

      10. associate-/l* N/A

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

      11. associate-*l* N/A

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

      12. sqrt-prod N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      13. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{{C}^{2}} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      16. unpow2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\sqrt{C \cdot C} \cdot \sqrt{\color{blue}{\frac{F}{A}} \cdot -16}}} \]

      17. rem-sqrt-square-rev N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      18. lower-unsound-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}}}} \]

      19. lower-fabs.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\color{blue}{\frac{F}{A} \cdot -16}}}} \]

      20. lower-unsound-sqrt.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

      22. lower-/.f64 15.4%

          \[\leadsto -0.25 \cdot \frac{1}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}} \]

   6. Applied rewrites 15.4%

      \[\leadsto -0.25 \cdot \frac{1}{\color{blue}{\frac{C}{\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}}}} \]

   if 3.1499999999999999e-279 < B < 430

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

   if 430 < B

   1. Initial program 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

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

Alternative 10: 56.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|B\right| \leq 3.15 \cdot 10^{-279}:\\ \;\;\;\;\left(\left(\left|\mathsf{max}\left(A, C\right)\right| \cdot \sqrt{\frac{F}{\mathsf{min}\left(A, C\right)} \cdot -16}\right) \cdot -0.25\right) \cdot \frac{1}{\mathsf{max}\left(A, C\right)}\\ \mathbf{elif}\;\left|B\right| \leq 430:\\ \;\;\;\;0.25 \cdot \frac{\sqrt{-16 \cdot \left(\mathsf{min}\left(A, C\right) \cdot F\right)}}{\mathsf{min}\left(A, C\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (if (<= (fabs B) 3.15e-279)
   (*
    (* (* (fabs (fmax A C)) (sqrt (* (/ F (fmin A C)) -16.0))) -0.25)
    (/ 1.0 (fmax A C)))
   (if (<= (fabs B) 430.0)
     (* 0.25 (/ (sqrt (* -16.0 (* (fmin A C) F))) (fmin A C)))
     (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B))))))))

C

double code(double A, double B, double C, double F) {
	double tmp;
	if (fabs(B) <= 3.15e-279) {
		tmp = ((fabs(fmax(A, C)) * sqrt(((F / fmin(A, C)) * -16.0))) * -0.25) * (1.0 / fmax(A, C));
	} else if (fabs(B) <= 430.0) {
		tmp = 0.25 * (sqrt((-16.0 * (fmin(A, C) * F))) / fmin(A, C));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

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) :: tmp
    if (abs(b) <= 3.15d-279) then
        tmp = ((abs(fmax(a, c)) * sqrt(((f / fmin(a, c)) * (-16.0d0)))) * (-0.25d0)) * (1.0d0 / fmax(a, c))
    else if (abs(b) <= 430.0d0) then
        tmp = 0.25d0 * (sqrt(((-16.0d0) * (fmin(a, c) * f))) / fmin(a, c))
    else
        tmp = (-1.0d0) * (sqrt(f) * sqrt((2.0d0 / abs(b))))
    end if
    code = tmp
end function

Java

public static double code(double A, double B, double C, double F) {
	double tmp;
	if (Math.abs(B) <= 3.15e-279) {
		tmp = ((Math.abs(fmax(A, C)) * Math.sqrt(((F / fmin(A, C)) * -16.0))) * -0.25) * (1.0 / fmax(A, C));
	} else if (Math.abs(B) <= 430.0) {
		tmp = 0.25 * (Math.sqrt((-16.0 * (fmin(A, C) * F))) / fmin(A, C));
	} else {
		tmp = -1.0 * (Math.sqrt(F) * Math.sqrt((2.0 / Math.abs(B))));
	}
	return tmp;
}

Python

def code(A, B, C, F):
	tmp = 0
	if math.fabs(B) <= 3.15e-279:
		tmp = ((math.fabs(fmax(A, C)) * math.sqrt(((F / fmin(A, C)) * -16.0))) * -0.25) * (1.0 / fmax(A, C))
	elif math.fabs(B) <= 430.0:
		tmp = 0.25 * (math.sqrt((-16.0 * (fmin(A, C) * F))) / fmin(A, C))
	else:
		tmp = -1.0 * (math.sqrt(F) * math.sqrt((2.0 / math.fabs(B))))
	return tmp

Julia

function code(A, B, C, F)
	tmp = 0.0
	if (abs(B) <= 3.15e-279)
		tmp = Float64(Float64(Float64(abs(fmax(A, C)) * sqrt(Float64(Float64(F / fmin(A, C)) * -16.0))) * -0.25) * Float64(1.0 / fmax(A, C)));
	elseif (abs(B) <= 430.0)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * Float64(fmin(A, C) * F))) / fmin(A, C)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (abs(B) <= 3.15e-279)
		tmp = ((abs(max(A, C)) * sqrt(((F / min(A, C)) * -16.0))) * -0.25) * (1.0 / max(A, C));
	elseif (abs(B) <= 430.0)
		tmp = 0.25 * (sqrt((-16.0 * (min(A, C) * F))) / min(A, C));
	else
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / abs(B))));
	end
	tmp_2 = tmp;
end

Wolfram

code[A_, B_, C_, F_] := If[LessEqual[N[Abs[B], $MachinePrecision], 3.15e-279], N[(N[(N[(N[Abs[N[Max[A, C], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] * N[(1.0 / N[Max[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[B], $MachinePrecision], 430.0], N[(0.25 * N[(N[Sqrt[N[(-16.0 * N[(N[Min[A, C], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.1499999999999999e-279

   1. Initial program 18.7%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

      7. lower-pow.f64 8.6%

         \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C} \]

   4. Applied rewrites 8.6%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{C}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{{C}^{2} \cdot F}{A}}}{\color{blue}{C}} \]

      4. mult-flip N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 15.3%

      \[\leadsto \left(\left(\left|C\right| \cdot \sqrt{\frac{F}{A} \cdot -16}\right) \cdot -0.25\right) \cdot \color{blue}{\frac{1}{C}} \]

   if 3.1499999999999999e-279 < B < 430

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

   if 430 < B

   1. Initial program 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

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

Alternative 11: 56.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|B\right| \leq 430:\\ \;\;\;\;0.25 \cdot \frac{\sqrt{-16 \cdot \left(\mathsf{min}\left(A, C\right) \cdot F\right)}}{\mathsf{min}\left(A, C\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{2}{\left|B\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (if (<= (fabs B) 430.0)
   (* 0.25 (/ (sqrt (* -16.0 (* (fmin A C) F))) (fmin A C)))
   (* -1.0 (* (sqrt F) (sqrt (/ 2.0 (fabs B)))))))

C

double code(double A, double B, double C, double F) {
	double tmp;
	if (fabs(B) <= 430.0) {
		tmp = 0.25 * (sqrt((-16.0 * (fmin(A, C) * F))) / fmin(A, C));
	} else {
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / fabs(B))));
	}
	return tmp;
}

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) :: tmp
    if (abs(b) <= 430.0d0) then
        tmp = 0.25d0 * (sqrt(((-16.0d0) * (fmin(a, c) * f))) / fmin(a, c))
    else
        tmp = (-1.0d0) * (sqrt(f) * sqrt((2.0d0 / abs(b))))
    end if
    code = tmp
end function

Java

public static double code(double A, double B, double C, double F) {
	double tmp;
	if (Math.abs(B) <= 430.0) {
		tmp = 0.25 * (Math.sqrt((-16.0 * (fmin(A, C) * F))) / fmin(A, C));
	} else {
		tmp = -1.0 * (Math.sqrt(F) * Math.sqrt((2.0 / Math.abs(B))));
	}
	return tmp;
}

Python

def code(A, B, C, F):
	tmp = 0
	if math.fabs(B) <= 430.0:
		tmp = 0.25 * (math.sqrt((-16.0 * (fmin(A, C) * F))) / fmin(A, C))
	else:
		tmp = -1.0 * (math.sqrt(F) * math.sqrt((2.0 / math.fabs(B))))
	return tmp

Julia

function code(A, B, C, F)
	tmp = 0.0
	if (abs(B) <= 430.0)
		tmp = Float64(0.25 * Float64(sqrt(Float64(-16.0 * Float64(fmin(A, C) * F))) / fmin(A, C)));
	else
		tmp = Float64(-1.0 * Float64(sqrt(F) * sqrt(Float64(2.0 / abs(B)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (abs(B) <= 430.0)
		tmp = 0.25 * (sqrt((-16.0 * (min(A, C) * F))) / min(A, C));
	else
		tmp = -1.0 * (sqrt(F) * sqrt((2.0 / abs(B))));
	end
	tmp_2 = tmp;
end

Wolfram

code[A_, B_, C_, F_] := If[LessEqual[N[Abs[B], $MachinePrecision], 430.0], N[(0.25 * N[(N[Sqrt[N[(-16.0 * N[(N[Min[A, C], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / N[Abs[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 430

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

   if 430 < B

   1. Initial program 18.7%

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

   3. Applied rewrites 21.8%

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

   4. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 18.8%

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

   6. Applied rewrites 18.8%

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

Alternative 12: 46.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\mathsf{min}\left(A, C\right) \leq -3.8 \cdot 10^{-159}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{\mathsf{min}\left(A, C\right)}}\right)\\ \mathbf{elif}\;\mathsf{min}\left(A, C\right) \leq 6.8 \cdot 10^{-261}:\\ \;\;\;\;-\sqrt{\left|-2 \cdot \frac{F}{B}\right|}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{\mathsf{min}\left(A, C\right)}}\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (if (<= (fmin A C) -3.8e-159)
   (* -0.25 (* (sqrt F) (sqrt (/ -16.0 (fmin A C)))))
   (if (<= (fmin A C) 6.8e-261)
     (- (sqrt (fabs (* -2.0 (/ F B)))))
     (* 0.25 (sqrt (* -16.0 (/ F (fmin A C))))))))

C

double code(double A, double B, double C, double F) {
	double tmp;
	if (fmin(A, C) <= -3.8e-159) {
		tmp = -0.25 * (sqrt(F) * sqrt((-16.0 / fmin(A, C))));
	} else if (fmin(A, C) <= 6.8e-261) {
		tmp = -sqrt(fabs((-2.0 * (F / B))));
	} else {
		tmp = 0.25 * sqrt((-16.0 * (F / fmin(A, C))));
	}
	return tmp;
}

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) :: tmp
    if (fmin(a, c) <= (-3.8d-159)) then
        tmp = (-0.25d0) * (sqrt(f) * sqrt(((-16.0d0) / fmin(a, c))))
    else if (fmin(a, c) <= 6.8d-261) then
        tmp = -sqrt(abs(((-2.0d0) * (f / b))))
    else
        tmp = 0.25d0 * sqrt(((-16.0d0) * (f / fmin(a, c))))
    end if
    code = tmp
end function

Java

public static double code(double A, double B, double C, double F) {
	double tmp;
	if (fmin(A, C) <= -3.8e-159) {
		tmp = -0.25 * (Math.sqrt(F) * Math.sqrt((-16.0 / fmin(A, C))));
	} else if (fmin(A, C) <= 6.8e-261) {
		tmp = -Math.sqrt(Math.abs((-2.0 * (F / B))));
	} else {
		tmp = 0.25 * Math.sqrt((-16.0 * (F / fmin(A, C))));
	}
	return tmp;
}

Python

def code(A, B, C, F):
	tmp = 0
	if fmin(A, C) <= -3.8e-159:
		tmp = -0.25 * (math.sqrt(F) * math.sqrt((-16.0 / fmin(A, C))))
	elif fmin(A, C) <= 6.8e-261:
		tmp = -math.sqrt(math.fabs((-2.0 * (F / B))))
	else:
		tmp = 0.25 * math.sqrt((-16.0 * (F / fmin(A, C))))
	return tmp

Julia

function code(A, B, C, F)
	tmp = 0.0
	if (fmin(A, C) <= -3.8e-159)
		tmp = Float64(-0.25 * Float64(sqrt(F) * sqrt(Float64(-16.0 / fmin(A, C)))));
	elseif (fmin(A, C) <= 6.8e-261)
		tmp = Float64(-sqrt(abs(Float64(-2.0 * Float64(F / B)))));
	else
		tmp = Float64(0.25 * sqrt(Float64(-16.0 * Float64(F / fmin(A, C)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (min(A, C) <= -3.8e-159)
		tmp = -0.25 * (sqrt(F) * sqrt((-16.0 / min(A, C))));
	elseif (min(A, C) <= 6.8e-261)
		tmp = -sqrt(abs((-2.0 * (F / B))));
	else
		tmp = 0.25 * sqrt((-16.0 * (F / min(A, C))));
	end
	tmp_2 = tmp;
end

Wolfram

code[A_, B_, C_, F_] := If[LessEqual[N[Min[A, C], $MachinePrecision], -3.8e-159], N[(-0.25 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(-16.0 / N[Min[A, C], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Min[A, C], $MachinePrecision], 6.8e-261], (-N[Sqrt[N[Abs[N[(-2.0 * N[(F / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), N[(0.25 * N[Sqrt[N[(-16.0 * N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if A < -3.8000000000000001e-159

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{\left(-16 \cdot A\right) \cdot F}}{A} \]

      5. sqrt-prod N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      6. lower-unsound-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      7. lower-unsound-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

      9. lower-unsound-sqrt.f64 18.0%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   6. Applied rewrites 18.0%

      \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot A} \cdot \sqrt{F}}{A} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{-16}{A}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

      5. lower-/.f64 18.0%

         \[\leadsto -0.25 \cdot \left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right) \]

   9. Applied rewrites 18.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{\frac{-16}{A}}\right)} \]

   if -3.8000000000000001e-159 < A < 6.8e-261

   1. Initial program 18.7%

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

   2. Taylor expanded in B around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 14.0%

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

   4. Applied rewrites 14.0%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 14.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 14.0%

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

   6. Applied rewrites 14.0%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 28.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 28.1%

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

   8. Applied rewrites 28.1%

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

   if 6.8e-261 < A

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

   5. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      3. lower-/.f64 10.4%

         \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

   7. Applied rewrites 10.4%

      \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 44.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{-16 \cdot \frac{F}{\mathsf{min}\left(A, C\right)}}\\ t_1 := -\sqrt{\left|-2 \cdot \frac{F}{B}\right|}\\ t_2 := {B}^{2} - \left(4 \cdot \mathsf{min}\left(A, C\right)\right) \cdot \mathsf{max}\left(A, C\right)\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(\mathsf{min}\left(A, C\right) + \mathsf{max}\left(A, C\right)\right) + \sqrt{{\left(\mathsf{min}\left(A, C\right) - \mathsf{max}\left(A, C\right)\right)}^{2} + {B}^{2}}\right)}}{t\_2}\\ t_4 := -0.25 \cdot t\_0\\ \mathbf{if}\;t\_3 \leq -6 \cdot 10^{+130}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1.34 \cdot 10^{-167}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;0.25 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* -16.0 (/ F (fmin A C)))))
        (t_1 (- (sqrt (fabs (* -2.0 (/ F B))))))
        (t_2 (- (pow B 2.0) (* (* 4.0 (fmin A C)) (fmax A C))))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+
              (+ (fmin A C) (fmax A C))
              (sqrt (+ (pow (- (fmin A C) (fmax A C)) 2.0) (pow B 2.0)))))))
          t_2))
        (t_4 (* -0.25 t_0)))
   (if (<= t_3 -6e+130)
     t_4
     (if (<= t_3 -1.34e-167)
       t_1
       (if (<= t_3 0.0) t_4 (if (<= t_3 INFINITY) (* 0.25 t_0) t_1))))))

C

double code(double A, double B, double C, double F) {
	double t_0 = sqrt((-16.0 * (F / fmin(A, C))));
	double t_1 = -sqrt(fabs((-2.0 * (F / B))));
	double t_2 = pow(B, 2.0) - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((fmin(A, C) + fmax(A, C)) + sqrt((pow((fmin(A, C) - fmax(A, C)), 2.0) + pow(B, 2.0)))))) / t_2;
	double t_4 = -0.25 * t_0;
	double tmp;
	if (t_3 <= -6e+130) {
		tmp = t_4;
	} else if (t_3 <= -1.34e-167) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = 0.25 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((-16.0 * (F / fmin(A, C))));
	double t_1 = -Math.sqrt(Math.abs((-2.0 * (F / B))));
	double t_2 = Math.pow(B, 2.0) - ((4.0 * fmin(A, C)) * fmax(A, C));
	double t_3 = -Math.sqrt(((2.0 * (t_2 * F)) * ((fmin(A, C) + fmax(A, C)) + Math.sqrt((Math.pow((fmin(A, C) - fmax(A, C)), 2.0) + Math.pow(B, 2.0)))))) / t_2;
	double t_4 = -0.25 * t_0;
	double tmp;
	if (t_3 <= -6e+130) {
		tmp = t_4;
	} else if (t_3 <= -1.34e-167) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = 0.25 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(A, B, C, F):
	t_0 = math.sqrt((-16.0 * (F / fmin(A, C))))
	t_1 = -math.sqrt(math.fabs((-2.0 * (F / B))))
	t_2 = math.pow(B, 2.0) - ((4.0 * fmin(A, C)) * fmax(A, C))
	t_3 = -math.sqrt(((2.0 * (t_2 * F)) * ((fmin(A, C) + fmax(A, C)) + math.sqrt((math.pow((fmin(A, C) - fmax(A, C)), 2.0) + math.pow(B, 2.0)))))) / t_2
	t_4 = -0.25 * t_0
	tmp = 0
	if t_3 <= -6e+130:
		tmp = t_4
	elif t_3 <= -1.34e-167:
		tmp = t_1
	elif t_3 <= 0.0:
		tmp = t_4
	elif t_3 <= math.inf:
		tmp = 0.25 * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(A, B, C, F)
	t_0 = sqrt(Float64(-16.0 * Float64(F / fmin(A, C))))
	t_1 = Float64(-sqrt(abs(Float64(-2.0 * Float64(F / B)))))
	t_2 = Float64((B ^ 2.0) - Float64(Float64(4.0 * fmin(A, C)) * fmax(A, C)))
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(fmin(A, C) + fmax(A, C)) + sqrt(Float64((Float64(fmin(A, C) - fmax(A, C)) ^ 2.0) + (B ^ 2.0))))))) / t_2)
	t_4 = Float64(-0.25 * t_0)
	tmp = 0.0
	if (t_3 <= -6e+130)
		tmp = t_4;
	elseif (t_3 <= -1.34e-167)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = Float64(0.25 * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((-16.0 * (F / min(A, C))));
	t_1 = -sqrt(abs((-2.0 * (F / B))));
	t_2 = (B ^ 2.0) - ((4.0 * min(A, C)) * max(A, C));
	t_3 = -sqrt(((2.0 * (t_2 * F)) * ((min(A, C) + max(A, C)) + sqrt((((min(A, C) - max(A, C)) ^ 2.0) + (B ^ 2.0)))))) / t_2;
	t_4 = -0.25 * t_0;
	tmp = 0.0;
	if (t_3 <= -6e+130)
		tmp = t_4;
	elseif (t_3 <= -1.34e-167)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = 0.25 * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(-16.0 * N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[N[Abs[N[(-2.0 * N[(F / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * N[Min[A, C], $MachinePrecision]), $MachinePrecision] * N[Max[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Min[A, C], $MachinePrecision] + N[Max[A, C], $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(N[Min[A, C], $MachinePrecision] - N[Max[A, C], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(-0.25 * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$3, -6e+130], t$95$4, If[LessEqual[t$95$3, -1.34e-167], t$95$1, If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, Infinity], N[(0.25 * t$95$0), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.9999999999999999e130 or -1.33999999999999998e-167 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\sqrt{-16 \cdot \frac{F}{A}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      4. lower-/.f64 14.7%

         \[\leadsto -0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

   7. Applied rewrites 14.7%

      \[\leadsto -0.25 \cdot \color{blue}{\sqrt{-16 \cdot \frac{F}{A}}} \]

   if -5.9999999999999999e130 < (/.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))) < -1.33999999999999998e-167 or +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 18.7%

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

   2. Taylor expanded in B around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 14.0%

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

   4. Applied rewrites 14.0%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 14.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 14.0%

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

   6. Applied rewrites 14.0%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 28.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 28.1%

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

   8. Applied rewrites 28.1%

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

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

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

   5. Taylor expanded in A around inf

      \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      3. lower-/.f64 10.4%

         \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

   7. Applied rewrites 10.4%

      \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 41.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|B\right| \leq 980:\\ \;\;\;\;-0.25 \cdot \sqrt{-16 \cdot \frac{F}{\mathsf{min}\left(A, C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\left|-2 \cdot \frac{F}{\left|B\right|}\right|}\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (if (<= (fabs B) 980.0)
   (* -0.25 (sqrt (* -16.0 (/ F (fmin A C)))))
   (- (sqrt (fabs (* -2.0 (/ F (fabs B))))))))

C

double code(double A, double B, double C, double F) {
	double tmp;
	if (fabs(B) <= 980.0) {
		tmp = -0.25 * sqrt((-16.0 * (F / fmin(A, C))));
	} else {
		tmp = -sqrt(fabs((-2.0 * (F / fabs(B)))));
	}
	return tmp;
}

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) :: tmp
    if (abs(b) <= 980.0d0) then
        tmp = (-0.25d0) * sqrt(((-16.0d0) * (f / fmin(a, c))))
    else
        tmp = -sqrt(abs(((-2.0d0) * (f / abs(b)))))
    end if
    code = tmp
end function

Java

public static double code(double A, double B, double C, double F) {
	double tmp;
	if (Math.abs(B) <= 980.0) {
		tmp = -0.25 * Math.sqrt((-16.0 * (F / fmin(A, C))));
	} else {
		tmp = -Math.sqrt(Math.abs((-2.0 * (F / Math.abs(B)))));
	}
	return tmp;
}

Python

def code(A, B, C, F):
	tmp = 0
	if math.fabs(B) <= 980.0:
		tmp = -0.25 * math.sqrt((-16.0 * (F / fmin(A, C))))
	else:
		tmp = -math.sqrt(math.fabs((-2.0 * (F / math.fabs(B)))))
	return tmp

Julia

function code(A, B, C, F)
	tmp = 0.0
	if (abs(B) <= 980.0)
		tmp = Float64(-0.25 * sqrt(Float64(-16.0 * Float64(F / fmin(A, C)))));
	else
		tmp = Float64(-sqrt(abs(Float64(-2.0 * Float64(F / abs(B))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (abs(B) <= 980.0)
		tmp = -0.25 * sqrt((-16.0 * (F / min(A, C))));
	else
		tmp = -sqrt(abs((-2.0 * (F / abs(B)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[A_, B_, C_, F_] := If[LessEqual[N[Abs[B], $MachinePrecision], 980.0], N[(-0.25 * N[Sqrt[N[(-16.0 * N[(F / N[Min[A, C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[Abs[N[(-2.0 * N[(F / N[Abs[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 980

   1. Initial program 18.7%

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

   2. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\color{blue}{A}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

      5. lower-*.f64 19.3%

         \[\leadsto 0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A} \]

   4. Applied rewrites 19.3%

      \[\leadsto \color{blue}{0.25 \cdot \frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{A}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\sqrt{-16 \cdot \frac{F}{A}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

      4. lower-/.f64 14.7%

         \[\leadsto -0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}} \]

   7. Applied rewrites 14.7%

      \[\leadsto -0.25 \cdot \color{blue}{\sqrt{-16 \cdot \frac{F}{A}}} \]

   if 980 < B

   1. Initial program 18.7%

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

   2. Taylor expanded in B around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 14.0%

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

   4. Applied rewrites 14.0%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 14.0%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 14.0%

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

   6. Applied rewrites 14.0%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-abs-rev N/A

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

      5. mul-fabs N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-fabs.f64 28.1%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 28.1%

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

   8. Applied rewrites 28.1%

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

Alternative 15: 28.1% accurate, 9.7× speedup

Math

\[-\sqrt{\left|-2 \cdot \frac{F}{B}\right|} \]

FPCore

(FPCore (A B C F) :precision binary64 (- (sqrt (fabs (* -2.0 (/ F B))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt(fabs((-2.0 * (F / B))));
}

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
    code = -sqrt(abs(((-2.0d0) * (f / b))))
end function

Java

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

Python

def code(A, B, C, F):
	return -math.sqrt(math.fabs((-2.0 * (F / B))))

Julia

function code(A, B, C, F)
	return Float64(-sqrt(abs(Float64(-2.0 * Float64(F / B)))))
end

MATLAB

function tmp = code(A, B, C, F)
	tmp = -sqrt(abs((-2.0 * (F / B))));
end

Wolfram

code[A_, B_, C_, F_] := (-N[Sqrt[N[Abs[N[(-2.0 * N[(F / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 18.7%

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

2. Taylor expanded in B around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 14.0%

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

4. Applied rewrites 14.0%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 14.0%

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 14.0%

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

6. Applied rewrites 14.0%

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

   1. rem-square-sqrt N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqr-abs-rev N/A

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

   5. mul-fabs N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-fabs.f64 28.1%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 28.1%

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

8. Applied rewrites 28.1%

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

Alternative 16: 14.0% accurate, 10.8× speedup

Math

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

FPCore

(FPCore (A B C F) :precision binary64 (- (sqrt (* F (/ -2.0 B)))))

C

double code(double A, double B, double C, double F) {
	return -sqrt((F * (-2.0 / B)));
}

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
    code = -sqrt((f * ((-2.0d0) / b)))
end function

Java

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

Python

def code(A, B, C, F):
	return -math.sqrt((F * (-2.0 / B)))

Julia

function code(A, B, C, F)
	return Float64(-sqrt(Float64(F * Float64(-2.0 / B))))
end

MATLAB

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

Wolfram

code[A_, B_, C_, F_] := (-N[Sqrt[N[(F * N[(-2.0 / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 18.7%

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

2. Taylor expanded in B around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 14.0%

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

4. Applied rewrites 14.0%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 14.0%

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 14.0%

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

6. Applied rewrites 14.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 14.0%

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

8. Applied rewrites 14.0%

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

Reproduce

herbie shell --seed 2025183 
(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))))