ABCF->ab-angle b

Percentage Accurate: 18.6% → 48.8%

Time: 20.5s

Alternatives: 13

Speedup: 11.7×

• could not determine a ground truth

  1. A = 5.892489202047567e-65
  2. B = 8.31579997374149e-19
  3. C = 1.689228959638423e+27
  4. F = -3.155569431825511e+290

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c, f)
    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: 48.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B}^{2}\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B 2.0)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_0) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          t_1)))
   (if (<= t_2 (- INFINITY))
     (*
      -0.25
      (* (/ (* (sqrt 2.0) (sqrt 2.0)) C) (* (sqrt (* F -2.0)) (sqrt (+ C C)))))
     (if (<= t_2 -1e-178)
       (/
        (*
         (sqrt (* 2.0 (fma B B (* (* A C) -4.0))))
         (sqrt (* F (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B)))))))
        t_1)
       (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0)))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B, 2.0);
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_0) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -0.25 * (((sqrt(2.0) * sqrt(2.0)) / C) * (sqrt((F * -2.0)) * sqrt((C + C))));
	} else if (t_2 <= -1e-178) {
		tmp = (sqrt((2.0 * fma(B, B, ((A * C) * -4.0)))) * sqrt((F * ((A + C) - sqrt(fma((A - C), (A - C), (B * B))))))) / t_1;
	} else {
		tmp = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B ^ 2.0))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-0.25 * Float64(Float64(Float64(sqrt(2.0) * sqrt(2.0)) / C) * Float64(sqrt(Float64(F * -2.0)) * sqrt(Float64(C + C)))));
	elseif (t_2 <= -1e-178)
		tmp = Float64(Float64(sqrt(Float64(2.0 * fma(B, B, Float64(Float64(A * C) * -4.0)))) * sqrt(Float64(F * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))))) / t_1);
	else
		tmp = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-0.25 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] * N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-178], N[(N[(N[Sqrt[N[(2.0 * N[(B * B + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.8%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 24.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 24.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. +-lowering-+.f64 32.3

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{\color{blue}{C + C}}\right)\right) \]

   9. Applied egg-rr 32.3%

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \color{blue}{\left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)}\right) \]

   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))) < -9.9999999999999995e-179

   1. Initial program 94.1%

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

      1. associate-*r* N/A

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

      2. associate-*l* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 95.7%

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

   if -9.9999999999999995e-179 < (/.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 4.5%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.7%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 19.7

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 19.7%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 19.9

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 19.9%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 19.9

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 19.9%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{t\_0 \cdot \left(\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 (- INFINITY))
     (*
      -0.25
      (* (/ (* (sqrt 2.0) (sqrt 2.0)) C) (* (sqrt (* F -2.0)) (sqrt (+ C C)))))
     (if (<= t_2 -1e-178)
       (*
        (sqrt
         (*
          t_0
          (* (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B)))) (* 2.0 F))))
        (/ -1.0 t_0))
       (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0)))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -0.25 * (((sqrt(2.0) * sqrt(2.0)) / C) * (sqrt((F * -2.0)) * sqrt((C + C))));
	} else if (t_2 <= -1e-178) {
		tmp = sqrt((t_0 * (((A + C) - sqrt(fma((A - C), (A - C), (B * B)))) * (2.0 * F)))) * (-1.0 / t_0);
	} else {
		tmp = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-0.25 * Float64(Float64(Float64(sqrt(2.0) * sqrt(2.0)) / C) * Float64(sqrt(Float64(F * -2.0)) * sqrt(Float64(C + C)))));
	elseif (t_2 <= -1e-178)
		tmp = Float64(sqrt(Float64(t_0 * Float64(Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) * Float64(2.0 * F)))) * Float64(-1.0 / t_0));
	else
		tmp = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-0.25 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] * N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-178], N[(N[Sqrt[N[(t$95$0 * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.8%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 24.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 24.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. +-lowering-+.f64 32.3

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{\color{blue}{C + C}}\right)\right) \]

   9. Applied egg-rr 32.3%

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \color{blue}{\left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)}\right) \]

   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))) < -9.9999999999999995e-179

   1. Initial program 94.1%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 62.5%

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

   6. Applied egg-rr 94.3%

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

   if -9.9999999999999995e-179 < (/.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 4.5%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.7%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 19.7

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 19.7%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 19.9

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 19.9%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 19.9

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 19.9%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 (- INFINITY))
     (*
      -0.25
      (* (/ (* (sqrt 2.0) (sqrt 2.0)) C) (* (sqrt (* F -2.0)) (sqrt (+ C C)))))
     (if (<= t_2 -1e-178)
       (/
        (sqrt
         (*
          t_0
          (* (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B)))) (* 2.0 F))))
        (- t_0))
       (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0)))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -0.25 * (((sqrt(2.0) * sqrt(2.0)) / C) * (sqrt((F * -2.0)) * sqrt((C + C))));
	} else if (t_2 <= -1e-178) {
		tmp = sqrt((t_0 * (((A + C) - sqrt(fma((A - C), (A - C), (B * B)))) * (2.0 * F)))) / -t_0;
	} else {
		tmp = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-0.25 * Float64(Float64(Float64(sqrt(2.0) * sqrt(2.0)) / C) * Float64(sqrt(Float64(F * -2.0)) * sqrt(Float64(C + C)))));
	elseif (t_2 <= -1e-178)
		tmp = Float64(sqrt(Float64(t_0 * Float64(Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) * Float64(2.0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-0.25 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] * N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-178], N[(N[Sqrt[N[(t$95$0 * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.8%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 24.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 24.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. +-lowering-+.f64 32.3

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{\color{blue}{C + C}}\right)\right) \]

   9. Applied egg-rr 32.3%

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \color{blue}{\left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)}\right) \]

   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))) < -9.9999999999999995e-179

   1. Initial program 94.1%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 62.5%

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

   6. Applied egg-rr 94.1%

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

   if -9.9999999999999995e-179 < (/.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 4.5%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.7%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 19.7

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 19.7%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 19.9

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 19.9%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 19.9

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 19.9%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* (* A C) -4.0)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 (- INFINITY))
     (*
      -0.25
      (* (/ (* (sqrt 2.0) (sqrt 2.0)) C) (* (sqrt (* F -2.0)) (sqrt (+ C C)))))
     (if (<= t_2 -1e-178)
       (/
        (sqrt
         (*
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))
          (* t_0 (* 2.0 F))))
        (- t_0))
       (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0)))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, ((A * C) * -4.0));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -0.25 * (((sqrt(2.0) * sqrt(2.0)) / C) * (sqrt((F * -2.0)) * sqrt((C + C))));
	} else if (t_2 <= -1e-178) {
		tmp = sqrt((((A + C) - sqrt(fma((A - C), (A - C), (B * B)))) * (t_0 * (2.0 * F)))) / -t_0;
	} else {
		tmp = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(Float64(A * C) * -4.0))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-0.25 * Float64(Float64(Float64(sqrt(2.0) * sqrt(2.0)) / C) * Float64(sqrt(Float64(F * -2.0)) * sqrt(Float64(C + C)))));
	elseif (t_2 <= -1e-178)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))) * Float64(t_0 * Float64(2.0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-0.25 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] * N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-178], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.8%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 24.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 24.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. +-lowering-+.f64 32.3

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{\color{blue}{C + C}}\right)\right) \]

   9. Applied egg-rr 32.3%

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \color{blue}{\left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)}\right) \]

   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))) < -9.9999999999999995e-179

   1. Initial program 94.1%

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

      1. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 94.1%

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

   if -9.9999999999999995e-179 < (/.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 4.5%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.7%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 19.7

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 19.7%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 19.9

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 19.9%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 19.9

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 19.9%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B}^{2}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+158}:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_0) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B 2.0)))))
   (if (<= t_1 -2e+158)
     (*
      -0.25
      (* (/ (* (sqrt 2.0) (sqrt 2.0)) C) (* (sqrt (* F -2.0)) (sqrt (+ C C)))))
     (if (<= t_1 -2e-147)
       (*
        (sqrt
         (/
          (* F (- (+ A C) (sqrt (fma B B (* (- A C) (- A C))))))
          (fma (* A C) -4.0 (* B B))))
        (- (sqrt 2.0)))
       (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0)))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B, 2.0) - t_0) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B, 2.0));
	double tmp;
	if (t_1 <= -2e+158) {
		tmp = -0.25 * (((sqrt(2.0) * sqrt(2.0)) / C) * (sqrt((F * -2.0)) * sqrt((C + C))));
	} else if (t_1 <= -2e-147) {
		tmp = sqrt(((F * ((A + C) - sqrt(fma(B, B, ((A - C) * (A - C)))))) / fma((A * C), -4.0, (B * B)))) * -sqrt(2.0);
	} else {
		tmp = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -2e+158)
		tmp = Float64(-0.25 * Float64(Float64(Float64(sqrt(2.0) * sqrt(2.0)) / C) * Float64(sqrt(Float64(F * -2.0)) * sqrt(Float64(C + C)))));
	elseif (t_1 <= -2e-147)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B, B, Float64(Float64(A - C) * Float64(A - C)))))) / fma(Float64(A * C), -4.0, Float64(B * B)))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+158], N[(-0.25 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] * N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-147], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B * B + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.99999999999999991e158

   1. Initial program 7.9%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 4.0%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 24.8

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 24.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. +-lowering-+.f64 32.6

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{\color{blue}{C + C}}\right)\right) \]

   9. Applied egg-rr 32.6%

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \color{blue}{\left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)}\right) \]

   if -1.99999999999999991e158 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-147

   1. Initial program 95.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. *-lowering-*.f64 N/A

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

   5. Simplified 97.4%

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

   if -1.9999999999999999e-147 < (/.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 5.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 1.5%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 19.9

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 19.9%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 20.1

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 20.1%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 20.1

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 20.1%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+158}:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B}^{2}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+212}:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_0) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B 2.0)))))
   (if (<= t_1 -1e+212)
     (*
      -0.25
      (* (/ (* (sqrt 2.0) (sqrt 2.0)) C) (* (sqrt (* F -2.0)) (sqrt (+ C C)))))
     (if (<= t_1 -2e-147)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma B B (* A A))))))) (- B))
       (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0)))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B, 2.0) - t_0) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B, 2.0));
	double tmp;
	if (t_1 <= -1e+212) {
		tmp = -0.25 * (((sqrt(2.0) * sqrt(2.0)) / C) * (sqrt((F * -2.0)) * sqrt((C + C))));
	} else if (t_1 <= -2e-147) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(B, B, (A * A))))))) / -B;
	} else {
		tmp = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -1e+212)
		tmp = Float64(-0.25 * Float64(Float64(Float64(sqrt(2.0) * sqrt(2.0)) / C) * Float64(sqrt(Float64(F * -2.0)) * sqrt(Float64(C + C)))));
	elseif (t_1 <= -2e-147)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A))))))) / Float64(-B));
	else
		tmp = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+212], N[(-0.25 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] * N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-147], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999991e211

   1. Initial program 4.9%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.8%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 25.4

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 25.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. +-lowering-+.f64 33.4

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{\color{blue}{C + C}}\right)\right) \]

   9. Applied egg-rr 33.4%

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \color{blue}{\left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)}\right) \]

   if -9.9999999999999991e211 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-147

   1. Initial program 95.7%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. /-lowering-/.f64 N/A

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

   5. Simplified 35.4%

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

   if -1.9999999999999999e-147 < (/.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 5.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 1.5%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 19.9

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 19.9%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 20.1

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 20.1%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 20.1

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 20.1%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{+212}:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -2} \cdot \sqrt{C + C}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B}^{2}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+212}:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -4} \cdot \sqrt{C}\right)\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_0) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B 2.0)))))
   (if (<= t_1 -1e+212)
     (*
      -0.25
      (* (/ (* (sqrt 2.0) (sqrt 2.0)) C) (* (sqrt (* F -4.0)) (sqrt C))))
     (if (<= t_1 -2e-147)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma B B (* A A))))))) (- B))
       (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0)))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B, 2.0) - t_0) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B, 2.0));
	double tmp;
	if (t_1 <= -1e+212) {
		tmp = -0.25 * (((sqrt(2.0) * sqrt(2.0)) / C) * (sqrt((F * -4.0)) * sqrt(C)));
	} else if (t_1 <= -2e-147) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(B, B, (A * A))))))) / -B;
	} else {
		tmp = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -1e+212)
		tmp = Float64(-0.25 * Float64(Float64(Float64(sqrt(2.0) * sqrt(2.0)) / C) * Float64(sqrt(Float64(F * -4.0)) * sqrt(C))));
	elseif (t_1 <= -2e-147)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A))))))) / Float64(-B));
	else
		tmp = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+212], N[(-0.25 * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] * N[(N[Sqrt[N[(F * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-147], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999991e211

   1. Initial program 4.9%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 0.8%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 25.4

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 25.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-out N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-lowering-sqrt.f64 33.4

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -4} \cdot \color{blue}{\sqrt{C}}\right)\right) \]

   9. Applied egg-rr 33.4%

      \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \color{blue}{\left(\sqrt{F \cdot -4} \cdot \sqrt{C}\right)}\right) \]

   if -9.9999999999999991e211 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-147

   1. Initial program 95.7%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. /-lowering-/.f64 N/A

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

   5. Simplified 35.4%

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

   if -1.9999999999999999e-147 < (/.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 5.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 1.5%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 19.9

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 19.9%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 20.1

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 20.1%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 20.1

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 20.1%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{+212}:\\ \;\;\;\;-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \left(\sqrt{F \cdot -4} \cdot \sqrt{C}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+212}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0))))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 -1e+212)
     t_0
     (if (<= t_2 -2e-147)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma B B (* A A))))))) (- B))
       t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -1e+212) {
		tmp = t_0;
	} else if (t_2 <= -2e-147) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(B, B, (A * A))))))) / -B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -1e+212)
		tmp = t_0;
	elseif (t_2 <= -2e-147)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A))))))) / Float64(-B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+212], t$95$0, If[LessEqual[t$95$2, -2e-147], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 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))) < -9.9999999999999991e211 or -1.9999999999999999e-147 < (/.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 5.2%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 1.4%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 21.4

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 21.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 21.6

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 21.6%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 21.6

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 21.6%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   if -9.9999999999999991e211 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-147

   1. Initial program 95.7%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. /-lowering-/.f64 N/A

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

   5. Simplified 35.4%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{+212}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+212}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-147}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0))))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 -1e+212)
     t_0
     (if (<= t_2 -2e-147)
       (- (* (/ (sqrt 2.0) B) (sqrt (* F (- A (sqrt (fma A A (* B B))))))))
       t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -1e+212) {
		tmp = t_0;
	} else if (t_2 <= -2e-147) {
		tmp = -((sqrt(2.0) / B) * sqrt((F * (A - sqrt(fma(A, A, (B * B)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -1e+212)
		tmp = t_0;
	elseif (t_2 <= -2e-147)
		tmp = Float64(-Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B))))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+212], t$95$0, If[LessEqual[t$95$2, -2e-147], (-N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), t$95$0]]]]]

Derivation

1. Split input into 2 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))) < -9.9999999999999991e211 or -1.9999999999999999e-147 < (/.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 5.2%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 1.4%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 21.4

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 21.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 21.6

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 21.6%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 21.6

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 21.6%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   if -9.9999999999999991e211 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e-147

   1. Initial program 95.7%

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 35.2

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

   7. Simplified 35.2%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{+212}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-147}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 2.55 \cdot 10^{-129}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A \cdot A\right) \cdot -16\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= A 2.55e-129)
   (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0)))))
   (/ (sqrt (* (* (* A A) -16.0) (* C F))) (- (fma A (* C -4.0) (* B B))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= 2.55e-129) {
		tmp = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
	} else {
		tmp = sqrt((((A * A) * -16.0) * (C * F))) / -fma(A, (C * -4.0), (B * B));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (A <= 2.55e-129)
		tmp = Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(A * A) * -16.0) * Float64(C * F))) / Float64(-fma(A, Float64(C * -4.0), Float64(B * B))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[A, 2.55e-129], N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A * A), $MachinePrecision] * -16.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 2.5499999999999999e-129

   1. Initial program 21.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-div N/A

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

      5. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 10.5%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-lowering-*.f64 29.3

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

   7. Simplified 29.3%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
   8. Step-by-step derivation

      1. rem-square-sqrt 29.7

         \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

   9. Applied egg-rr 29.7%

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
   10. Step-by-step derivation

       1. distribute-rgt-out N/A

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

       2. *-lowering-*.f64 N/A

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

       3. metadata-eval 29.7

          \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

   11. Applied egg-rr 29.7%

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   if 2.5499999999999999e-129 < A

   1. Initial program 7.2%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. +-lowering-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 7.1

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

   5. Simplified 7.1%

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

   7. Applied egg-rr 7.1%

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

   8. Taylor expanded in B around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 6.4

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

   10. Simplified 6.4%

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

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 2.55 \cdot 10^{-129}:\\ \;\;\;\;-0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A \cdot A\right) \cdot -16\right) \cdot \left(C \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right) \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (* -0.25 (* (/ 2.0 C) (sqrt (* F (* C -4.0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-0.25d0) * ((2.0d0 / c) * sqrt((f * (c * (-4.0d0)))))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return -0.25 * ((2.0 / C) * Math.sqrt((F * (C * -4.0))));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return -0.25 * ((2.0 / C) * math.sqrt((F * (C * -4.0))))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(-0.25 * Float64(Float64(2.0 / C) * sqrt(Float64(F * Float64(C * -4.0)))))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = -0.25 * ((2.0 / C) * sqrt((F * (C * -4.0))));
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-0.25 * N[(N[(2.0 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

   1. *-commutative N/A

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

   2. flip-- N/A

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

   3. associate-*l/ N/A

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

   4. sqrt-div N/A

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

   5. /-lowering-/.f64 N/A

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

4. Applied egg-rr 9.2%

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

5. Taylor expanded in A around -inf

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

   1. *-lowering-*.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. *-lowering-*.f64 20.3

       \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

7. Simplified 20.3%

   \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
8. Step-by-step derivation

   1. rem-square-sqrt 20.6

      \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]

9. Applied egg-rr 20.6%

   \[\leadsto -0.25 \cdot \left(\frac{\color{blue}{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right) \]
10. Step-by-step derivation

    1. distribute-rgt-out N/A

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

    2. *-lowering-*.f64 N/A

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

    3. metadata-eval 20.6

       \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \left(C \cdot \color{blue}{-4}\right)}\right) \]

11. Applied egg-rr 20.6%

    \[\leadsto -0.25 \cdot \left(\frac{2}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]
12. Add Preprocessing

Alternative 12: 3.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{-0.5}{C} \cdot \sqrt{F \cdot \left(A + A\right)} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* (/ -0.5 C) (sqrt (* F (+ A A)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return (-0.5 / C) * sqrt((F * (A + A)));
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = ((-0.5d0) / c) * sqrt((f * (a + a)))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return (-0.5 / C) * Math.sqrt((F * (A + A)));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return (-0.5 / C) * math.sqrt((F * (A + A)))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(Float64(-0.5 / C) * sqrt(Float64(F * Float64(A + A))))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = (-0.5 / C) * sqrt((F * (A + A)));
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[(-0.5 / C), $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

   1. *-commutative N/A

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

   2. flip-- N/A

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

   3. associate-*l/ N/A

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

   4. sqrt-div N/A

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

   5. /-lowering-/.f64 N/A

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

4. Applied egg-rr 9.2%

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

5. Taylor expanded in A around -inf

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

   1. *-lowering-*.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. *-lowering-*.f64 20.3

       \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, \color{blue}{-2 \cdot C}\right)}\right) \]

7. Simplified 20.3%

   \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-2, C, -2 \cdot C\right)}\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

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

   8. flip-+ N/A

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

   9. +-inverses N/A

      \[\leadsto \frac{\frac{-1}{2}}{C} \cdot \sqrt{F \cdot \frac{\left(-2 \cdot C\right) \cdot \left(-2 \cdot C\right) - \left(-2 \cdot C\right) \cdot \left(-2 \cdot C\right)}{\color{blue}{0}}} \]

   10. +-inverses N/A

       \[\leadsto \frac{\frac{-1}{2}}{C} \cdot \sqrt{F \cdot \frac{\color{blue}{0}}{0}} \]

   11. +-inverses N/A

       \[\leadsto \frac{\frac{-1}{2}}{C} \cdot \sqrt{F \cdot \frac{\color{blue}{A \cdot A - A \cdot A}}{0}} \]

   12. +-inverses N/A

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

   13. flip-+ N/A

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

   14. *-lowering-*.f64 N/A

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

   15. +-lowering-+.f64 2.2

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

9. Applied egg-rr 2.2%

   \[\leadsto \color{blue}{\frac{-0.5}{C} \cdot \sqrt{F \cdot \left(A + A\right)}} \]
10. Add Preprocessing

Alternative 13: 2.0% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt(((2.0d0 * f) / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. neg-lowering-neg.f64 N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sqrt-lowering-sqrt.f64 1.9

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

5. Simplified 1.9%

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

   1. mul-1-neg N/A

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

   2. remove-double-neg N/A

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

   3. sqrt-lowering-sqrt.f64 1.9

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

7. Applied egg-rr 1.9%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-commutative N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-lowering-*.f64 1.9

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

9. Applied egg-rr 1.9%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))