ABCF->ab-angle b

Percentage Accurate: 18.8% → 53.2%

Time: 14.6s

Alternatives: 12

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 18.8% 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: 53.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_0}{t\_1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\frac{-1}{\frac{t\_1}{\sqrt{t\_0 \cdot \left(\left(2 \cdot F\right) \cdot t\_1\right)}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_4 \cdot F\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)}}{-t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (+ C A) (hypot (- A C) B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (* F (/ t_0 t_1))) (- (sqrt 2.0)))
     (if (<= t_3 -5e-187)
       (/ -1.0 (/ t_1 (sqrt (* t_0 (* (* 2.0 F) t_1)))))
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* (* (* t_4 F) 2.0) (+ (fma (/ (* B_m B_m) C) -0.5 A) A)))
          (- t_4))
         (/ (sqrt (* 2.0 (* (- A (hypot A B_m)) F))) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (C + A) - hypot((A - C), B_m);
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt((F * (t_0 / t_1))) * -sqrt(2.0);
	} else if (t_3 <= -5e-187) {
		tmp = -1.0 / (t_1 / sqrt((t_0 * ((2.0 * F) * t_1))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((((t_4 * F) * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A))) / -t_4;
	} else {
		tmp = sqrt((2.0 * ((A - hypot(A, B_m)) * F))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(C + A) - hypot(Float64(A - C), B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(t_0 / t_1))) * Float64(-sqrt(2.0)));
	elseif (t_3 <= -5e-187)
		tmp = Float64(-1.0 / Float64(t_1 / sqrt(Float64(t_0 * Float64(Float64(2.0 * F) * t_1)))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_4 * F) * 2.0) * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))) / Float64(-t_4));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(A - hypot(A, B_m)) * F))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(F * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -5e-187], N[(-1.0 / N[(t$95$1 / N[Sqrt[N[(t$95$0 * N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(N[(t$95$4 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.1%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 61.6%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999996e-187

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 98.7%

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

   if -4.9999999999999996e-187 < (/.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 23.0%

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

   4. Applied rewrites 33.6%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 32.7

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{C}, A\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 32.7%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 32.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 19.5

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

   5. Applied rewrites 19.5%

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

   7. Applied rewrites 19.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_2 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_0}{t\_1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\left(2 \cdot F\right) \cdot t\_1\right)}}{-t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_2 \cdot F\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)}}{-t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (+ C A) (hypot (- A C) B_m)))
        (t_1 (fma -4.0 (* C A) (* B_m B_m)))
        (t_2 (fma (* C A) -4.0 (* B_m B_m)))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3))))
   (if (<= t_4 (- INFINITY))
     (* (sqrt (* F (/ t_0 t_1))) (- (sqrt 2.0)))
     (if (<= t_4 -5e-187)
       (/ (sqrt (* t_0 (* (* 2.0 F) t_1))) (- t_1))
       (if (<= t_4 INFINITY)
         (/
          (sqrt (* (* (* t_2 F) 2.0) (+ (fma (/ (* B_m B_m) C) -0.5 A) A)))
          (- t_2))
         (/ (sqrt (* 2.0 (* (- A (hypot A B_m)) F))) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (C + A) - hypot((A - C), B_m);
	double t_1 = fma(-4.0, (C * A), (B_m * B_m));
	double t_2 = fma((C * A), -4.0, (B_m * B_m));
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = sqrt((F * (t_0 / t_1))) * -sqrt(2.0);
	} else if (t_4 <= -5e-187) {
		tmp = sqrt((t_0 * ((2.0 * F) * t_1))) / -t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((((t_2 * F) * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A))) / -t_2;
	} else {
		tmp = sqrt((2.0 * ((A - hypot(A, B_m)) * F))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(C + A) - hypot(Float64(A - C), B_m))
	t_1 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_2 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_3))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(t_0 / t_1))) * Float64(-sqrt(2.0)));
	elseif (t_4 <= -5e-187)
		tmp = Float64(sqrt(Float64(t_0 * Float64(Float64(2.0 * F) * t_1))) / Float64(-t_1));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_2 * F) * 2.0) * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))) / Float64(-t_2));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(A - hypot(A, B_m)) * F))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[Sqrt[N[(F * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, -5e-187], N[(N[Sqrt[N[(t$95$0 * N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(N[(t$95$2 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.1%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 61.6%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999996e-187

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 98.7%

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

   if -4.9999999999999996e-187 < (/.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 23.0%

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

   4. Applied rewrites 33.6%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 32.7

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{C}, A\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 32.7%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 32.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 19.5

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

   5. Applied rewrites 19.5%

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

   7. Applied rewrites 19.5%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+304}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e-85)
     (*
      (sqrt
       (*
        (fma (* C A) -4.0 (* B_m B_m))
        (* (* F 2.0) (+ (fma (/ (* B_m B_m) C) -0.5 A) A))))
      (/ -1.0 t_0))
     (if (<= (pow B_m 2.0) 1e+304)
       (* (sqrt (* F (/ (- (+ C A) (hypot (- A C) B_m)) t_0))) (- (sqrt 2.0)))
       (/ (sqrt (* 2.0 (* (- A (hypot A B_m)) F))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-85) {
		tmp = sqrt((fma((C * A), -4.0, (B_m * B_m)) * ((F * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A)))) * (-1.0 / t_0);
	} else if (pow(B_m, 2.0) <= 1e+304) {
		tmp = sqrt((F * (((C + A) - hypot((A - C), B_m)) / t_0))) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 * ((A - hypot(A, B_m)) * F))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-85)
		tmp = Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * Float64(Float64(F * 2.0) * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A)))) * Float64(-1.0 / t_0));
	elseif ((B_m ^ 2.0) <= 1e+304)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / t_0))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(A - hypot(A, B_m)) * F))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-85], N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+304], N[(N[Sqrt[N[(F * N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e-85

   1. Initial program 27.9%

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 28.3

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{C}, A\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 28.3%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   9. Applied rewrites 29.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 2e-85 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999994e303

   1. Initial program 38.7%

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

   3. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 65.5%

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

   if 9.9999999999999994e303 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 29.3

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

   5. Applied rewrites 29.3%

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

   7. Applied rewrites 29.4%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+304}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-85)
   (*
    (sqrt
     (*
      (fma (* C A) -4.0 (* B_m B_m))
      (* (* F 2.0) (+ (fma (/ (* B_m B_m) C) -0.5 A) A))))
    (/ -1.0 (fma -4.0 (* C A) (* B_m B_m))))
   (/ (sqrt (* 2.0 (* (- A (hypot A B_m)) F))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-85) {
		tmp = sqrt((fma((C * A), -4.0, (B_m * B_m)) * ((F * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A)))) * (-1.0 / fma(-4.0, (C * A), (B_m * B_m)));
	} else {
		tmp = sqrt((2.0 * ((A - hypot(A, B_m)) * F))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-85)
		tmp = Float64(sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * Float64(Float64(F * 2.0) * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A)))) * Float64(-1.0 / fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(A - hypot(A, B_m)) * F))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-85], N[(N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e-85

   1. Initial program 27.9%

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 28.3

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{C}, A\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 28.3%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

   9. Applied rewrites 29.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 2e-85 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 20.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 26.3

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

   5. Applied rewrites 26.3%

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

   7. Applied rewrites 26.3%

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

Alternative 5: 43.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+16}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot 2\right) \cdot \left(\left(t\_0 \cdot F\right) \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, B\_m \cdot F, 2 \cdot \left(A \cdot F\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e+16)
     (/ (sqrt (* (* A 2.0) (* (* t_0 F) 2.0))) (- t_0))
     (/ (sqrt (fma -2.0 (* B_m F) (* 2.0 (* A F)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+16) {
		tmp = sqrt(((A * 2.0) * ((t_0 * F) * 2.0))) / -t_0;
	} else {
		tmp = sqrt(fma(-2.0, (B_m * F), (2.0 * (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+16)
		tmp = Float64(sqrt(Float64(Float64(A * 2.0) * Float64(Float64(t_0 * F) * 2.0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(B_m * F), Float64(2.0 * Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+16], N[(N[Sqrt[N[(N[(A * 2.0), $MachinePrecision] * N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e16

   1. Initial program 31.4%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 30.9

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

   5. Applied rewrites 30.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 30.9%

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

   if 1e16 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.9%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 25.2

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

   5. Applied rewrites 25.2%

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

   7. Applied rewrites 25.3%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 21.7%

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

Alternative 6: 38.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5.6 \cdot 10^{-14}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \mathbf{elif}\;F \leq -5.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, B\_m \cdot F, 2 \cdot \left(A \cdot F\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5.6e-14)
   (- (sqrt (* -2.0 (/ F B_m))))
   (if (<= F -5.4e-303)
     (/ (sqrt (fma -2.0 (* B_m F) (* 2.0 (* A F)))) (- B_m))
     (*
      (sqrt (* (* -16.0 (* A A)) (* C F)))
      (/ -1.0 (fma -4.0 (* C A) (* B_m B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5.6e-14) {
		tmp = -sqrt((-2.0 * (F / B_m)));
	} else if (F <= -5.4e-303) {
		tmp = sqrt(fma(-2.0, (B_m * F), (2.0 * (A * F)))) / -B_m;
	} else {
		tmp = sqrt(((-16.0 * (A * A)) * (C * F))) * (-1.0 / fma(-4.0, (C * A), (B_m * B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -5.6e-14)
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	elseif (F <= -5.4e-303)
		tmp = Float64(sqrt(fma(-2.0, Float64(B_m * F), Float64(2.0 * Float64(A * F)))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(Float64(-16.0 * Float64(A * A)) * Float64(C * F))) * Float64(-1.0 / fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5.6e-14], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[F, -5.4e-303], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.6000000000000001e-14

   1. Initial program 22.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.6%

      \[\leadsto \frac{0 - {\left(\sqrt{\frac{F}{B} \cdot 2} \cdot -1\right)}^{2}}{\color{blue}{0 + \sqrt{\frac{F}{B} \cdot 2} \cdot -1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 0.8%

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

   11. Applied rewrites 17.9%

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

   if -5.6000000000000001e-14 < F < -5.39999999999999972e-303

   1. Initial program 22.3%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 30.2

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

   5. Applied rewrites 30.2%

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

   7. Applied rewrites 30.4%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 23.7%

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

   if -5.39999999999999972e-303 < F

   1. Initial program 32.3%

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

   4. Applied rewrites 48.5%

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

   5. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(-16 \cdot {A}^{2}\right)} \cdot \left(C \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 17.5

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

   7. Applied rewrites 17.5%

      \[\leadsto \sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(C \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 41.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, B\_m \cdot F, 2 \cdot \left(A \cdot F\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.1e-36)
   (*
    (sqrt (* -8.0 (* A (* C (* F (+ A A))))))
    (/ -1.0 (fma -4.0 (* C A) (* B_m B_m))))
   (/ (sqrt (fma -2.0 (* B_m F) (* 2.0 (* A F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.1e-36) {
		tmp = sqrt((-8.0 * (A * (C * (F * (A + A)))))) * (-1.0 / fma(-4.0, (C * A), (B_m * B_m)));
	} else {
		tmp = sqrt(fma(-2.0, (B_m * F), (2.0 * (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.1e-36)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A)))))) * Float64(-1.0 / fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(B_m * F), Float64(2.0 * Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.1e-36], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.09999999999999991e-36

   1. Initial program 23.5%

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

   4. Applied rewrites 28.3%

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

   5. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 19.0

         \[\leadsto \sqrt{\left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{C}, A\right) + A\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 19.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)} \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 18.1

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

   10. Applied rewrites 18.1%

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

   if 2.09999999999999991e-36 < B

   1. Initial program 25.1%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 52.5

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

   5. Applied rewrites 52.5%

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

   7. Applied rewrites 52.6%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 43.5%

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

Alternative 8: 35.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5.6 \cdot 10^{-14}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, B\_m \cdot F, 2 \cdot \left(A \cdot F\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5.6e-14)
   (- (sqrt (* -2.0 (/ F B_m))))
   (/ (sqrt (fma -2.0 (* B_m F) (* 2.0 (* A F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5.6e-14) {
		tmp = -sqrt((-2.0 * (F / B_m)));
	} else {
		tmp = sqrt(fma(-2.0, (B_m * F), (2.0 * (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -5.6e-14)
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(B_m * F), Float64(2.0 * Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5.6e-14], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.6000000000000001e-14

   1. Initial program 22.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.6%

      \[\leadsto \frac{0 - {\left(\sqrt{\frac{F}{B} \cdot 2} \cdot -1\right)}^{2}}{\color{blue}{0 + \sqrt{\frac{F}{B} \cdot 2} \cdot -1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 0.8%

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

   11. Applied rewrites 17.9%

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

   if -5.6000000000000001e-14 < F

   1. Initial program 24.9%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 23.2

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

   5. Applied rewrites 23.2%

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

   7. Applied rewrites 23.3%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 18.6%

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

Alternative 9: 35.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5.6 \cdot 10^{-14}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m} \cdot \sqrt{\mathsf{fma}\left(-B\_m, F, F \cdot A\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5.6e-14)
   (- (sqrt (* -2.0 (/ F B_m))))
   (* (/ -1.0 B_m) (sqrt (* (fma (- B_m) F (* F A)) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5.6e-14) {
		tmp = -sqrt((-2.0 * (F / B_m)));
	} else {
		tmp = (-1.0 / B_m) * sqrt((fma(-B_m, F, (F * A)) * 2.0));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -5.6e-14)
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	else
		tmp = Float64(Float64(-1.0 / B_m) * sqrt(Float64(fma(Float64(-B_m), F, Float64(F * A)) * 2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5.6e-14], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(N[((-B$95$m) * F + N[(F * A), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.6000000000000001e-14

   1. Initial program 22.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.6%

      \[\leadsto \frac{0 - {\left(\sqrt{\frac{F}{B} \cdot 2} \cdot -1\right)}^{2}}{\color{blue}{0 + \sqrt{\frac{F}{B} \cdot 2} \cdot -1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 0.8%

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

   11. Applied rewrites 17.9%

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

   if -5.6000000000000001e-14 < F

   1. Initial program 24.9%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 23.2

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

   5. Applied rewrites 23.2%

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

   7. Applied rewrites 23.3%

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

   9. Applied rewrites 23.2%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 18.6%

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

Alternative 10: 35.0% accurate, 12.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-45}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -2e-45)
   (- (sqrt (* -2.0 (/ F B_m))))
   (/ (sqrt (* -2.0 (* B_m F))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2e-45) {
		tmp = -sqrt((-2.0 * (F / B_m)));
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-2d-45)) then
        tmp = -sqrt(((-2.0d0) * (f / b_m)))
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / -b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2e-45) {
		tmp = -Math.sqrt((-2.0 * (F / B_m)));
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -2e-45:
		tmp = -math.sqrt((-2.0 * (F / B_m)))
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2e-45)
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -2e-45)
		tmp = -sqrt((-2.0 * (F / B_m)));
	else
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2e-45], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.99999999999999997e-45

   1. Initial program 21.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. 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. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 0.9

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

   5. Applied rewrites 0.9%

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

   7. Applied rewrites 0.8%

      \[\leadsto \frac{0 - {\left(\sqrt{\frac{F}{B} \cdot 2} \cdot -1\right)}^{2}}{\color{blue}{0 + \sqrt{\frac{F}{B} \cdot 2} \cdot -1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 0.9%

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

   11. Applied rewrites 18.0%

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

   if -1.99999999999999997e-45 < F

   1. Initial program 26.2%

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

   3. Taylor expanded in C around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 22.0

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

   5. Applied rewrites 22.0%

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 19.3%

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

Alternative 11: 28.1% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((-2.0d0) * (f / b_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in B around -inf

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

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 2.0

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

5. Applied rewrites 2.0%

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

7. Applied rewrites 1.0%

   \[\leadsto \frac{0 - {\left(\sqrt{\frac{F}{B} \cdot 2} \cdot -1\right)}^{2}}{\color{blue}{0 + \sqrt{\frac{F}{B} \cdot 2} \cdot -1}} \]
8. Step-by-step derivation

9. Applied rewrites 2.0%

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

11. Applied rewrites 13.3%

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

Alternative 12: 2.4% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((((-2.0d0) / b_m) * f))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

3. Taylor expanded in B around -inf

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

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-/.f64 2.0

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

5. Applied rewrites 2.0%

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

7. Applied rewrites 1.0%

   \[\leadsto \frac{0 - {\left(\sqrt{\frac{F}{B} \cdot 2} \cdot -1\right)}^{2}}{\color{blue}{0 + \sqrt{\frac{F}{B} \cdot 2} \cdot -1}} \]
8. Step-by-step derivation

9. Applied rewrites 2.0%

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

11. Applied rewrites 2.2%

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

Reproduce

herbie shell --seed 2024313 
(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))))