ABCF->ab-angle a

Percentage Accurate: 19.7% → 63.9%

Time: 14.6s

Alternatives: 16

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: 19.7% 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: 63.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_0}\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot t\_2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_0)))
        (t_2 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_1 -5e-210)
     (*
      (* (sqrt (* F 2.0)) (- (sqrt (fma (* C A) -4.0 (* B_m B_m)))))
      (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) t_2))
     (if (<= t_1 INFINITY)
       (*
        (/ (sqrt (* (* 2.0 F) t_2)) -1.0)
        (/ (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_2))
       (/ (* (sqrt F) (sqrt (* (+ (hypot C B_m) C) 2.0))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_0;
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_1 <= -5e-210) {
		tmp = (sqrt((F * 2.0)) * -sqrt(fma((C * A), -4.0, (B_m * B_m)))) * (sqrt(((hypot((A - C), B_m) + A) + C)) / t_2);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt(((2.0 * F) * t_2)) / -1.0) * (sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) / t_2);
	} else {
		tmp = (sqrt(F) * sqrt(((hypot(C, B_m) + C) * 2.0))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_0))
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_1 <= -5e-210)
		tmp = Float64(Float64(sqrt(Float64(F * 2.0)) * Float64(-sqrt(fma(Float64(C * A), -4.0, Float64(B_m * B_m))))) * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / t_2));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * t_2)) / -1.0) * Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) / t_2));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-210], N[(N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000002e-210

   1. Initial program 42.1%

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

   4. Applied rewrites 60.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. pow1/2 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow-prod-down N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. pow1/2 N/A

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

      18. lift-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. metadata-eval N/A

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

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

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

   6. Applied rewrites 79.4%

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

   if -5.0000000000000002e-210 < (/.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 20.1%

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

   4. Applied rewrites 36.1%

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

   5. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.8

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

   7. Applied rewrites 30.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 23.0

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

   5. Applied rewrites 23.0%

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

   7. Applied rewrites 30.5%

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

   9. Applied rewrites 30.6%

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

   11. Applied rewrites 30.6%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\left(\sqrt{F \cdot 2} \cdot \left(-\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\right)\right) \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-195}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(F \cdot 2\right)}}{\frac{-t\_1}{\sqrt{C + C}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot t\_0}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-195)
     (/ (sqrt (* t_1 (* F 2.0))) (/ (- t_1) (sqrt (+ C C))))
     (if (<= (pow B_m 2.0) 4e+56)
       (*
        (/ (sqrt (* (* 2.0 F) t_0)) -1.0)
        (/ (sqrt (+ (+ (hypot (- A C) B_m) A) C)) t_0))
       (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-195) {
		tmp = sqrt((t_1 * (F * 2.0))) / (-t_1 / sqrt((C + C)));
	} else if (pow(B_m, 2.0) <= 4e+56) {
		tmp = (sqrt(((2.0 * F) * t_0)) / -1.0) * (sqrt(((hypot((A - C), B_m) + A) + C)) / t_0);
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-195)
		tmp = Float64(sqrt(Float64(t_1 * Float64(F * 2.0))) / Float64(Float64(-t_1) / sqrt(Float64(C + C))));
	elseif ((B_m ^ 2.0) <= 4e+56)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) / -1.0) * Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) / t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-195], N[(N[Sqrt[N[(t$95$1 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-t$95$1) / N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+56], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.0000000000000001e-195

   1. Initial program 15.3%

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

   4. Applied rewrites 30.0%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 16.2

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 16.2%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

   9. Applied rewrites 20.2%

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

   if 1.0000000000000001e-195 < (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000037e56

   1. Initial program 42.9%

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

   4. Applied rewrites 53.9%

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

   if 4.00000000000000037e56 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 29.1

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

   5. Applied rewrites 29.1%

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

   7. Applied rewrites 37.5%

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

   9. Applied rewrites 37.6%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\frac{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{C + C}}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+47}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot 2\right)}}{\frac{-t\_0}{\sqrt{C + C}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

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) 1e+47)
     (/ (sqrt (* t_0 (* F 2.0))) (/ (- t_0) (sqrt (+ C C))))
     (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+47) {
		tmp = sqrt((t_0 * (F * 2.0))) / (-t_0 / sqrt((C + C)));
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+47)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * 2.0))) / Float64(Float64(-t_0) / sqrt(Float64(C + C))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+47], N[(N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-t$95$0) / N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e47

   1. Initial program 25.9%

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 20.6

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 20.6%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

   9. Applied rewrites 24.8%

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

   if 1e47 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.7%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 29.6

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

   5. Applied rewrites 29.6%

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

   7. Applied rewrites 38.0%

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

   9. Applied rewrites 38.1%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\frac{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{C + C}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 6.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot 2\right)}}{\frac{-t\_0}{\sqrt{C + C}}}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2} \cdot \frac{\sqrt{F}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \sqrt{{B\_m}^{-1}} \cdot \left(C \cdot \sqrt{2}\right), \sqrt{B\_m} \cdot \sqrt{2}\right)}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 6.2e+23)
     (/ (sqrt (* t_0 (* F 2.0))) (/ (- t_0) (sqrt (+ C C))))
     (if (<= B_m 1.05e+189)
       (* (sqrt (* (+ (hypot C B_m) C) 2.0)) (/ (sqrt F) (- B_m)))
       (*
        (/
         (fma
          0.5
          (* (sqrt (pow B_m -1.0)) (* C (sqrt 2.0)))
          (* (sqrt B_m) (sqrt 2.0)))
         B_m)
        (- (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 6.2e+23) {
		tmp = sqrt((t_0 * (F * 2.0))) / (-t_0 / sqrt((C + C)));
	} else if (B_m <= 1.05e+189) {
		tmp = sqrt(((hypot(C, B_m) + C) * 2.0)) * (sqrt(F) / -B_m);
	} else {
		tmp = (fma(0.5, (sqrt(pow(B_m, -1.0)) * (C * sqrt(2.0))), (sqrt(B_m) * sqrt(2.0))) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 6.2e+23)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * 2.0))) / Float64(Float64(-t_0) / sqrt(Float64(C + C))));
	elseif (B_m <= 1.05e+189)
		tmp = Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) * Float64(sqrt(F) / Float64(-B_m)));
	else
		tmp = Float64(Float64(fma(0.5, Float64(sqrt((B_m ^ -1.0)) * Float64(C * sqrt(2.0))), Float64(sqrt(B_m) * sqrt(2.0))) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6.2e+23], N[(N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-t$95$0) / N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.05e+189], N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(N[Sqrt[N[Power[B$95$m, -1.0], $MachinePrecision]], $MachinePrecision] * N[(C * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.19999999999999941e23

   1. Initial program 20.7%

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 15.0

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 15.0%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

   9. Applied rewrites 17.8%

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

   if 6.19999999999999941e23 < B < 1.04999999999999996e189

   1. Initial program 24.6%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 47.7

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 56.3%

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

   9. Applied rewrites 56.3%

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

   11. Applied rewrites 56.4%

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

   if 1.04999999999999996e189 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 67.4

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

   5. Applied rewrites 67.4%

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

   7. Applied rewrites 92.9%

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

   9. Applied rewrites 93.2%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 87.1%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{B}} \cdot \left(C \cdot \sqrt{2}\right), \sqrt{B} \cdot \sqrt{2}\right)}{-B} \cdot \sqrt{F} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\frac{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{C + C}}}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2} \cdot \frac{\sqrt{F}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \sqrt{{B}^{-1}} \cdot \left(C \cdot \sqrt{2}\right), \sqrt{B} \cdot \sqrt{2}\right)}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 7 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot 2\right)}}{\frac{-t\_0}{\sqrt{C + C}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \sqrt{{B\_m}^{-1}} \cdot \left(C \cdot \sqrt{2}\right), \sqrt{B\_m} \cdot \sqrt{2}\right)}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 7e+24)
     (/ (sqrt (* t_0 (* F 2.0))) (/ (- t_0) (sqrt (+ C C))))
     (*
      (/
       (fma
        0.5
        (* (sqrt (pow B_m -1.0)) (* C (sqrt 2.0)))
        (* (sqrt B_m) (sqrt 2.0)))
       B_m)
      (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 7e+24) {
		tmp = sqrt((t_0 * (F * 2.0))) / (-t_0 / sqrt((C + C)));
	} else {
		tmp = (fma(0.5, (sqrt(pow(B_m, -1.0)) * (C * sqrt(2.0))), (sqrt(B_m) * sqrt(2.0))) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 7e+24)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * 2.0))) / Float64(Float64(-t_0) / sqrt(Float64(C + C))));
	else
		tmp = Float64(Float64(fma(0.5, Float64(sqrt((B_m ^ -1.0)) * Float64(C * sqrt(2.0))), Float64(sqrt(B_m) * sqrt(2.0))) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 7e+24], N[(N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-t$95$0) / N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(N[Sqrt[N[Power[B$95$m, -1.0], $MachinePrecision]], $MachinePrecision] * N[(C * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 7.0000000000000004e24

   1. Initial program 20.7%

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 15.0

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 15.0%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

   9. Applied rewrites 17.8%

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

   if 7.0000000000000004e24 < B

   1. Initial program 13.1%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 73.4%

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

   9. Applied rewrites 73.6%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 69.4%

       \[\leadsto \frac{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{B}} \cdot \left(C \cdot \sqrt{2}\right), \sqrt{B} \cdot \sqrt{2}\right)}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\frac{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{C + C}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \sqrt{{B}^{-1}} \cdot \left(C \cdot \sqrt{2}\right), \sqrt{B} \cdot \sqrt{2}\right)}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+47}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot 2\right)}}{\frac{-t\_0}{\sqrt{C + C}}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e+47)
     (/ (sqrt (* t_0 (* F 2.0))) (/ (- t_0) (sqrt (+ C C))))
     (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+47) {
		tmp = sqrt((t_0 * (F * 2.0))) / (-t_0 / sqrt((C + C)));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+47)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * 2.0))) / Float64(Float64(-t_0) / sqrt(Float64(C + C))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+47], N[(N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-t$95$0) / N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e47

   1. Initial program 25.9%

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 20.6

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 20.6%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}}} \]

   9. Applied rewrites 24.8%

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

   if 1e47 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.7%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 27.9

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

   5. Applied rewrites 27.9%

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 34.5%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)}}{\frac{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}{\sqrt{C + C}}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+47}:\\ \;\;\;\;\frac{\sqrt{C + C}}{-t\_0} \cdot \sqrt{t\_0 \cdot \left(F \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e+47)
     (* (/ (sqrt (+ C C)) (- t_0)) (sqrt (* t_0 (* F 2.0))))
     (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+47) {
		tmp = (sqrt((C + C)) / -t_0) * sqrt((t_0 * (F * 2.0)));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+47)
		tmp = Float64(Float64(sqrt(Float64(C + C)) / Float64(-t_0)) * sqrt(Float64(t_0 * Float64(F * 2.0))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+47], N[(N[(N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision] * N[Sqrt[N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e47

   1. Initial program 25.9%

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 20.6

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 20.6%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}} \]

      3. lower-*.f64 20.6

         \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1}} \]

   9. Applied rewrites 24.8%

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

   if 1e47 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.7%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 27.9

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

   5. Applied rewrites 27.9%

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 34.5%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+47}:\\ \;\;\;\;\frac{\sqrt{C + C}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e+47)
     (/ (sqrt (* (+ C C) (* t_0 (* F 2.0)))) (- t_0))
     (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+47) {
		tmp = sqrt(((C + C) * (t_0 * (F * 2.0)))) / -t_0;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+47)
		tmp = Float64(sqrt(Float64(Float64(C + C) * Float64(t_0 * Float64(F * 2.0)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+47], N[(N[Sqrt[N[(N[(C + C), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e47

   1. Initial program 25.9%

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 20.6

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 20.6%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 23.3%

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

   if 1e47 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.7%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 27.9

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

   5. Applied rewrites 27.9%

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

   7. Applied rewrites 28.1%

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

   9. Applied rewrites 34.5%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+47}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-35)
   (/ (sqrt (* -16.0 (* A (* (* C C) F)))) (- (fma (* -4.0 C) A (* B_m B_m))))
   (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-35) {
		tmp = sqrt((-16.0 * (A * ((C * C) * F)))) / -fma((-4.0 * C), A, (B_m * B_m));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-35)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(Float64(C * C) * F)))) / Float64(-fma(Float64(-4.0 * C), A, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-35], N[(N[Sqrt[N[(-16.0 * N[(A * N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999964e-35

   1. Initial program 23.7%

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

   4. Applied rewrites 38.0%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 19.4

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 19.4%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 21.4%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 13.5

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

   12. Applied rewrites 13.5%

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

   if 4.99999999999999964e-35 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 25.1

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

   5. Applied rewrites 25.1%

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

   7. Applied rewrites 25.3%

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

   9. Applied rewrites 30.9%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 7 \cdot 10^{+25}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(C + C\right)\right) \cdot \left(F \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= B_m 2e-104)
     (/ (sqrt (* (+ C C) (* t_0 (* F 2.0)))) (* (* 4.0 A) C))
     (if (<= B_m 7e+25)
       (/ (sqrt (* (* t_0 (+ C C)) (* F 2.0))) (- t_0))
       (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (B_m <= 2e-104) {
		tmp = sqrt(((C + C) * (t_0 * (F * 2.0)))) / ((4.0 * A) * C);
	} else if (B_m <= 7e+25) {
		tmp = sqrt(((t_0 * (C + C)) * (F * 2.0))) / -t_0;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 2e-104)
		tmp = Float64(sqrt(Float64(Float64(C + C) * Float64(t_0 * Float64(F * 2.0)))) / Float64(Float64(4.0 * A) * C));
	elseif (B_m <= 7e+25)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(C + C)) * Float64(F * 2.0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2e-104], N[(N[Sqrt[N[(N[(C + C), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7e+25], N[(N[Sqrt[N[(N[(t$95$0 * N[(C + C), $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.99999999999999985e-104

   1. Initial program 17.4%

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

   4. Applied rewrites 28.0%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 14.1

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 14.1%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 16.3%

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

   10. Taylor expanded in A around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 11.6

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

   12. Applied rewrites 11.6%

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

   if 1.99999999999999985e-104 < B < 6.99999999999999999e25

   1. Initial program 40.2%

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 20.0

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 20.0%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 20.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 23.7

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

   11. Applied rewrites 23.7%

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

   if 6.99999999999999999e25 < B

   1. Initial program 13.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 55.0%

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

   9. Applied rewrites 69.1%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+25}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(C + C\right)\right) \cdot \left(F \cdot 2\right)}}{-\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot \left(F \cdot 2\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6.2e+23)
   (/
    (sqrt (* (+ C C) (* (fma (* -4.0 C) A (* B_m B_m)) (* F 2.0))))
    (* (* 4.0 A) C))
   (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6.2e+23) {
		tmp = sqrt(((C + C) * (fma((-4.0 * C), A, (B_m * B_m)) * (F * 2.0)))) / ((4.0 * A) * C);
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6.2e+23)
		tmp = Float64(sqrt(Float64(Float64(C + C) * Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * Float64(F * 2.0)))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6.2e+23], N[(N[Sqrt[N[(N[(C + C), $MachinePrecision] * N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.19999999999999941e23

   1. Initial program 20.7%

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

   4. Applied rewrites 31.0%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \left(\color{blue}{0} \cdot \frac{A}{C} + 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \color{blue}{\mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

      6. lower-/.f64 15.0

         \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{C \cdot \mathsf{fma}\left(0, \color{blue}{\frac{A}{C}}, 1\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   7. Applied rewrites 15.0%

      \[\leadsto \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\color{blue}{C \cdot \mathsf{fma}\left(0, \frac{A}{C}, 1\right)} + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   9. Applied rewrites 16.9%

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

   10. Taylor expanded in A around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 12.7

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

   12. Applied rewrites 12.7%

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

   if 6.19999999999999941e23 < B

   1. Initial program 13.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 55.0%

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

   9. Applied rewrites 69.1%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\left(C + C\right) \cdot \left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.3% accurate, 12.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.65 \cdot 10^{-254}:\\ \;\;\;\;\frac{2}{-B\_m} \cdot \sqrt{C \cdot F}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.65e-254)
   (* (/ 2.0 (- B_m)) (sqrt (* C F)))
   (- (sqrt (* (/ F B_m) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.65e-254) {
		tmp = (2.0 / -B_m) * sqrt((C * F));
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	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 <= 1.65d-254) then
        tmp = (2.0d0 / -b_m) * sqrt((c * f))
    else
        tmp = -sqrt(((f / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.65e-254) {
		tmp = (2.0 / -B_m) * Math.sqrt((C * F));
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.65e-254:
		tmp = (2.0 / -B_m) * math.sqrt((C * F))
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.65e-254)
		tmp = Float64(Float64(2.0 / Float64(-B_m)) * sqrt(Float64(C * F)));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.65e-254)
		tmp = (2.0 / -B_m) * sqrt((C * F));
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.65e-254], N[(N[(2.0 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 1.65000000000000008e-254

   1. Initial program 31.2%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 8.5

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

   5. Applied rewrites 8.5%

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

   7. Applied rewrites 8.2%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 3.6%

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

   if 1.65000000000000008e-254 < F

   1. Initial program 16.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 19.9

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

   5. Applied rewrites 19.9%

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

   7. Applied rewrites 20.0%

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

4. Final simplification 17.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.65 \cdot 10^{-254}:\\ \;\;\;\;\frac{2}{-B} \cdot \sqrt{C \cdot F}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.3% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (/ (- (sqrt F)) (sqrt (* B_m 0.5))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt(F) / sqrt((B_m * 0.5));
}

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(f) / sqrt((b_m * 0.5d0))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(F) / Math.sqrt((B_m * 0.5));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(F) / math.sqrt((B_m * 0.5))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(F) / sqrt((B_m * 0.5));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 16.6

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

5. Applied rewrites 16.6%

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

7. Applied rewrites 16.7%

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

9. Applied rewrites 16.7%

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

11. Applied rewrites 20.7%

    \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]

12. Final simplification 20.7%

    \[\leadsto \frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
13. Add Preprocessing

Alternative 14: 35.3% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = 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(f) * sqrt((2.0d0 / b_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 16.6

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

5. Applied rewrites 16.6%

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

7. Applied rewrites 16.7%

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

9. Applied rewrites 20.6%

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

10. Final simplification 20.6%

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

Alternative 15: 27.1% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = 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(((f / b_m) * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 16.6

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

5. Applied rewrites 16.6%

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

7. Applied rewrites 16.7%

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

Alternative 16: 27.1% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = 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((f * (2.0d0 / b_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 16.6

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

5. Applied rewrites 16.6%

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

7. Applied rewrites 16.7%

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

9. Applied rewrites 16.7%

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

Reproduce

herbie shell --seed 2024298 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))