ABCF->ab-angle a

Percentage Accurate: 18.9% → 53.3%

Time: 18.7s

Alternatives: 12

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 53.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+302}:\\ \;\;\;\;\sqrt{\frac{F}{\sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)} - A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 5e-157)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e+302)
       (* (sqrt (/ F (- (sqrt (fma A A (* B_m B_m))) A))) (- (sqrt 2.0)))
       (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 5e-157) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e+302) {
		tmp = sqrt((F / (sqrt(fma(A, A, (B_m * B_m))) - A))) * -sqrt(2.0);
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-157)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e+302)
		tmp = Float64(sqrt(Float64(F / Float64(sqrt(fma(A, A, Float64(B_m * B_m))) - A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(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 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-157], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+302], N[(N[Sqrt[N[(F / N[(N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000002e-157

   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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 24.0

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

   5. Applied rewrites 24.0%

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

   if 5.0000000000000002e-157 < (pow.f64 B #s(literal 2 binary64)) < 1.0000000000000001e302

   1. Initial program 26.8%

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

   4. Applied rewrites 26.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 20.6%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 44.6

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

   9. Applied rewrites 44.6%

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

   if 1.0000000000000001e302 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.0%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 18.9

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

   5. Applied rewrites 18.9%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 27.3

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

   7. Applied rewrites 27.3%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+302}:\\ \;\;\;\;\sqrt{\frac{F}{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)} - A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 44.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}} \leq 10^{+214}:\\ \;\;\;\;\sqrt{\frac{F}{\sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)} - A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<=
        (/
         (sqrt
          (*
           (* 2.0 (* (- (pow B_m 2.0) t_0) F))
           (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
         (- t_0 (pow B_m 2.0)))
        1e+214)
     (* (sqrt (/ F (- (sqrt (fma A A (* B_m B_m))) A))) (- (sqrt 2.0)))
     (/ (- (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) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if ((sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0))) <= 1e+214) {
		tmp = sqrt((F / (sqrt(fma(A, A, (B_m * B_m))) - A))) * -sqrt(2.0);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0))) <= 1e+214)
		tmp = Float64(sqrt(Float64(F / Float64(sqrt(fma(A, A, Float64(B_m * B_m))) - A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+214], N[(N[Sqrt[N[(F / N[(N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 9.9999999999999995e213

   1. Initial program 34.3%

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

   4. Applied rewrites 34.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 16.4%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 36.7

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

   9. Applied rewrites 36.7%

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

   if 9.9999999999999995e213 < (/.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 0.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 12.0

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

   5. Applied rewrites 12.0%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 17.4

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

   7. Applied rewrites 17.4%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 17.3

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

   9. Applied rewrites 17.3%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. distribute-lft-neg-in N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. metadata-eval N/A

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

       9. sqrt-div N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. un-div-inv N/A

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

       13. lower-/.f64 N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-sqrt.f64 N/A

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

       16. div-inv N/A

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

       17. metadata-eval N/A

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

       18. lower-*.f64 17.4

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

   11. Applied rewrites 17.4%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+214}:\\ \;\;\;\;\sqrt{\frac{F}{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)} - A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(2 \cdot C\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+302}:\\ \;\;\;\;\sqrt{\frac{F}{\sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)} - A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 B_m B_m (* -4.0 (* A C)))))
   (if (<= (pow B_m 2.0) 5e-157)
     (/ -1.0 (/ t_0 (sqrt (* (* 2.0 C) (* t_0 (* 2.0 F))))))
     (if (<= (pow B_m 2.0) 1e+302)
       (* (sqrt (/ F (- (sqrt (fma A A (* B_m B_m))) A))) (- (sqrt 2.0)))
       (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-157) {
		tmp = -1.0 / (t_0 / sqrt(((2.0 * C) * (t_0 * (2.0 * F)))));
	} else if (pow(B_m, 2.0) <= 1e+302) {
		tmp = sqrt((F / (sqrt(fma(A, A, (B_m * B_m))) - A))) * -sqrt(2.0);
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(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(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-157)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(Float64(2.0 * C) * Float64(t_0 * Float64(2.0 * F))))));
	elseif ((B_m ^ 2.0) <= 1e+302)
		tmp = Float64(sqrt(Float64(F / Float64(sqrt(fma(A, A, Float64(B_m * B_m))) - A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(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[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-157], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+302], N[(N[Sqrt[N[(F / N[(N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000002e-157

   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 23.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 24.0

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

   7. Applied rewrites 24.0%

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

   if 5.0000000000000002e-157 < (pow.f64 B #s(literal 2 binary64)) < 1.0000000000000001e302

   1. Initial program 26.8%

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

   4. Applied rewrites 26.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 20.6%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 44.6

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

   9. Applied rewrites 44.6%

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

   if 1.0000000000000001e302 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.0%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 18.9

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

   5. Applied rewrites 18.9%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 27.3

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

   7. Applied rewrites 27.3%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\left(2 \cdot C\right) \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+302}:\\ \;\;\;\;\sqrt{\frac{F}{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)} - A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 3 \cdot 10^{+128}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C}}{B\_m} \cdot \left(-\frac{\sqrt{F}}{0.5}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 3e+128)
   (/ (- (sqrt F)) (sqrt (* B_m 0.5)))
   (* (/ (sqrt C) B_m) (- (/ (sqrt F) 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) {
	double tmp;
	if (C <= 3e+128) {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	} else {
		tmp = (sqrt(C) / B_m) * -(sqrt(F) / 0.5);
	}
	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 (c <= 3d+128) then
        tmp = -sqrt(f) / sqrt((b_m * 0.5d0))
    else
        tmp = (sqrt(c) / b_m) * -(sqrt(f) / 0.5d0)
    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 (C <= 3e+128) {
		tmp = -Math.sqrt(F) / Math.sqrt((B_m * 0.5));
	} else {
		tmp = (Math.sqrt(C) / B_m) * -(Math.sqrt(F) / 0.5);
	}
	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 C <= 3e+128:
		tmp = -math.sqrt(F) / math.sqrt((B_m * 0.5))
	else:
		tmp = (math.sqrt(C) / B_m) * -(math.sqrt(F) / 0.5)
	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 (C <= 3e+128)
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	else
		tmp = Float64(Float64(sqrt(C) / B_m) * Float64(-Float64(sqrt(F) / 0.5)));
	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 (C <= 3e+128)
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	else
		tmp = (sqrt(C) / B_m) * -(sqrt(F) / 0.5);
	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[C, 3e+128], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[C], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[(N[Sqrt[F], $MachinePrecision] / 0.5), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 2.9999999999999998e128

   1. Initial program 20.9%

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

   3. Taylor expanded in B around inf

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 18.9

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

   7. Applied rewrites 18.9%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 18.9

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

   9. Applied rewrites 18.9%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. distribute-lft-neg-in N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. metadata-eval N/A

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

       9. sqrt-div N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. un-div-inv N/A

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

       13. lower-/.f64 N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-sqrt.f64 N/A

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

       16. div-inv N/A

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

       17. metadata-eval N/A

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

       18. lower-*.f64 18.9

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

   11. Applied rewrites 18.9%

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

   if 2.9999999999999998e128 < C

   1. Initial program 7.2%

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

   4. Applied rewrites 7.2%

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

   5. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 1.4

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

   7. Applied rewrites 1.4%

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

   8. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 4.9

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

   10. Applied rewrites 4.9%

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

       1. rem-square-sqrt N/A

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

       2. div-inv N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 N/A

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

       5. div-inv N/A

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

       6. clear-num N/A

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

       7. metadata-eval N/A

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

       8. associate-*l/ N/A

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

       9. div-inv N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. *-lft-identity N/A

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

       13. lift-sqrt.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. sqrt-prod N/A

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

       16. pow1/2 N/A

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

       17. lift-sqrt.f64 N/A

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

       18. times-frac N/A

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

       19. lower-*.f64 N/A

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

       20. lower-/.f64 N/A

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

       21. pow1/2 N/A

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

       22. lower-sqrt.f64 N/A

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

       23. lower-/.f64 10.8

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

   12. Applied rewrites 10.8%

       \[\leadsto -\color{blue}{\frac{\sqrt{C}}{B} \cdot \frac{\sqrt{F}}{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3 \cdot 10^{+128}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C}}{B} \cdot \left(-\frac{\sqrt{F}}{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 3 \cdot 10^{+128}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{C} \cdot \frac{-2}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 3e+128)
   (/ (- (sqrt F)) (sqrt (* B_m 0.5)))
   (* (sqrt F) (* (sqrt 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 tmp;
	if (C <= 3e+128) {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	} else {
		tmp = sqrt(F) * (sqrt(C) * (-2.0 / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 3d+128) then
        tmp = -sqrt(f) / sqrt((b_m * 0.5d0))
    else
        tmp = sqrt(f) * (sqrt(c) * ((-2.0d0) / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 3e+128) {
		tmp = -Math.sqrt(F) / Math.sqrt((B_m * 0.5));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt(C) * (-2.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 3e+128:
		tmp = -math.sqrt(F) / math.sqrt((B_m * 0.5))
	else:
		tmp = math.sqrt(F) * (math.sqrt(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)
	tmp = 0.0
	if (C <= 3e+128)
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(C) * Float64(-2.0 / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 3e+128)
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	else
		tmp = sqrt(F) * (sqrt(C) * (-2.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 3e+128], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 2.9999999999999998e128

   1. Initial program 20.9%

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

   3. Taylor expanded in B around inf

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 18.9

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

   7. Applied rewrites 18.9%

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

      1. lift-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lower-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 18.9

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

   9. Applied rewrites 18.9%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. distribute-lft-neg-in N/A

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

       5. lift-sqrt.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. clear-num N/A

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

       8. metadata-eval N/A

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

       9. sqrt-div N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

       12. un-div-inv N/A

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

       13. lower-/.f64 N/A

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

       14. lower-neg.f64 N/A

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

       15. lower-sqrt.f64 N/A

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

       16. div-inv N/A

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

       17. metadata-eval N/A

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

       18. lower-*.f64 18.9

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

   11. Applied rewrites 18.9%

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

   if 2.9999999999999998e128 < C

   1. Initial program 7.2%

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

   4. Applied rewrites 7.2%

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

   5. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 1.4

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

   7. Applied rewrites 1.4%

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

   8. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 4.9

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

   10. Applied rewrites 4.9%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. distribute-lft-neg-in N/A

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

       8. lift-sqrt.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. sqrt-prod N/A

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

       11. pow1/2 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. associate-*r* N/A

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

       14. lower-*.f64 N/A

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

   12. Applied rewrites 10.8%

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

4. Final simplification 17.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3 \cdot 10^{+128}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{C} \cdot \frac{-2}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{B\_m}{2 \cdot F}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 2.55e+92)
   (/ -1.0 (sqrt (/ B_m (* 2.0 F))))
   (/ (* -2.0 (sqrt (* C F))) B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 2.55e+92) {
		tmp = -1.0 / sqrt((B_m / (2.0 * F)));
	} else {
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 2.55d+92) then
        tmp = (-1.0d0) / sqrt((b_m / (2.0d0 * f)))
    else
        tmp = ((-2.0d0) * sqrt((c * f))) / b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 2.55e+92) {
		tmp = -1.0 / Math.sqrt((B_m / (2.0 * F)));
	} else {
		tmp = (-2.0 * Math.sqrt((C * F))) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 2.55e+92:
		tmp = -1.0 / math.sqrt((B_m / (2.0 * F)))
	else:
		tmp = (-2.0 * math.sqrt((C * F))) / B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 2.55e+92)
		tmp = Float64(-1.0 / sqrt(Float64(B_m / Float64(2.0 * F))));
	else
		tmp = Float64(Float64(-2.0 * sqrt(Float64(C * F))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 2.55e+92)
		tmp = -1.0 / sqrt((B_m / (2.0 * F)));
	else
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 2.55e+92], N[(-1.0 / N[Sqrt[N[(B$95$m / N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 2.5500000000000001e92

   1. Initial program 20.2%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 16.0

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

   5. Applied rewrites 16.0%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. pow1/2 N/A

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

      9. lower-/.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 19.4

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

   7. Applied rewrites 19.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 16.0

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

   9. Applied rewrites 16.0%

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

   if 2.5500000000000001e92 < C

   1. Initial program 13.2%

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

   4. Applied rewrites 13.1%

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

   5. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 1.4

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

   7. Applied rewrites 1.4%

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

   8. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 4.5

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

   10. Applied rewrites 4.5%

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

       1. rem-square-sqrt N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. associate-*l/ N/A

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

       5. distribute-neg-frac N/A

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

       6. lower-/.f64 N/A

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

       7. *-commutative N/A

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

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

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. metadata-eval 4.6

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

   12. Applied rewrites 4.6%

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

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{B}{2 \cdot F}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.5% 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 19.0%

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

3. Taylor expanded in 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

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 14.2

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

5. Applied rewrites 14.2%

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

   1. lift-/.f64 N/A

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

   2. sqrt-unprod N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. sqrt-div N/A

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

   8. pow1/2 N/A

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

   9. lower-/.f64 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 17.0

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

7. Applied rewrites 17.0%

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

   1. lift-*.f64 N/A

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

   2. sqrt-div N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. sqrt-prod N/A

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

   7. pow1/2 N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 17.0

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

9. Applied rewrites 17.0%

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

    1. lift-sqrt.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-sqrt.f64 N/A

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

    4. distribute-lft-neg-in N/A

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

    5. lift-sqrt.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. clear-num N/A

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

    8. metadata-eval N/A

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

    9. sqrt-div N/A

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

    10. metadata-eval N/A

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

    11. metadata-eval N/A

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

    12. un-div-inv N/A

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

    13. lower-/.f64 N/A

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

    14. lower-neg.f64 N/A

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

    15. lower-sqrt.f64 N/A

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

    16. div-inv N/A

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

    17. metadata-eval N/A

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

    18. lower-*.f64 17.0

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

11. Applied rewrites 17.0%

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

Alternative 8: 35.5% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.0%

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

3. Taylor expanded in 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

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 14.2

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

5. Applied rewrites 14.2%

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

   1. lift-/.f64 N/A

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

   2. sqrt-unprod N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. sqrt-div N/A

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

   8. pow1/2 N/A

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

   9. lower-/.f64 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 17.0

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

7. Applied rewrites 17.0%

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

Alternative 9: 35.5% 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 19.0%

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

3. Taylor expanded in 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

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 14.2

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

5. Applied rewrites 14.2%

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

   1. lift-/.f64 N/A

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

   2. sqrt-unprod N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. sqrt-div N/A

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

   8. pow1/2 N/A

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

   9. lower-/.f64 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 17.0

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

7. Applied rewrites 17.0%

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

   1. lift-*.f64 N/A

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

   2. sqrt-div N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. sqrt-prod N/A

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

   7. pow1/2 N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 17.0

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

9. Applied rewrites 17.0%

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

10. Final simplification 17.0%

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

Alternative 10: 27.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 2.55e+92)
   (- (sqrt (* F (/ 2.0 B_m))))
   (/ (* -2.0 (sqrt (* C F))) B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 2.55e+92) {
		tmp = -sqrt((F * (2.0 / B_m)));
	} else {
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 2.55d+92) then
        tmp = -sqrt((f * (2.0d0 / b_m)))
    else
        tmp = ((-2.0d0) * sqrt((c * f))) / b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 2.55e+92) {
		tmp = -Math.sqrt((F * (2.0 / B_m)));
	} else {
		tmp = (-2.0 * Math.sqrt((C * F))) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 2.55e+92:
		tmp = -math.sqrt((F * (2.0 / B_m)))
	else:
		tmp = (-2.0 * math.sqrt((C * F))) / B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 2.55e+92)
		tmp = Float64(-sqrt(Float64(F * Float64(2.0 / B_m))));
	else
		tmp = Float64(Float64(-2.0 * sqrt(Float64(C * F))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 2.55e+92)
		tmp = -sqrt((F * (2.0 / B_m)));
	else
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 2.55e+92], (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(-2.0 * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 2.5500000000000001e92

   1. Initial program 20.2%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 16.0

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

   5. Applied rewrites 16.0%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 16.0

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 16.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 16.0

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

   7. Applied rewrites 16.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 16.0

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

   9. Applied rewrites 16.0%

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

   if 2.5500000000000001e92 < C

   1. Initial program 13.2%

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

   4. Applied rewrites 13.1%

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

   5. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 1.4

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

   7. Applied rewrites 1.4%

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

   8. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 4.5

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

   10. Applied rewrites 4.5%

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

       1. rem-square-sqrt N/A

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

       2. lift-*.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. associate-*l/ N/A

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

       5. distribute-neg-frac N/A

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

       6. lower-/.f64 N/A

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

       7. *-commutative N/A

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

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

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

       9. lower-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. metadata-eval 4.6

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

   12. Applied rewrites 4.6%

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

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.6% accurate, 12.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B\_m} \cdot \sqrt{C \cdot F}\\ \end{array} \end{array} \]

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
 (if (<= C 2.55e+92)
   (- (sqrt (* F (/ 2.0 B_m))))
   (* (/ -2.0 B_m) (sqrt (* C 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 tmp;
	if (C <= 2.55e+92) {
		tmp = -sqrt((F * (2.0 / B_m)));
	} else {
		tmp = (-2.0 / B_m) * sqrt((C * F));
	}
	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 (c <= 2.55d+92) then
        tmp = -sqrt((f * (2.0d0 / b_m)))
    else
        tmp = ((-2.0d0) / b_m) * sqrt((c * f))
    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 (C <= 2.55e+92) {
		tmp = -Math.sqrt((F * (2.0 / B_m)));
	} else {
		tmp = (-2.0 / B_m) * Math.sqrt((C * F));
	}
	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 C <= 2.55e+92:
		tmp = -math.sqrt((F * (2.0 / B_m)))
	else:
		tmp = (-2.0 / B_m) * math.sqrt((C * 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)
	tmp = 0.0
	if (C <= 2.55e+92)
		tmp = Float64(-sqrt(Float64(F * Float64(2.0 / B_m))));
	else
		tmp = Float64(Float64(-2.0 / B_m) * sqrt(Float64(C * F)));
	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 (C <= 2.55e+92)
		tmp = -sqrt((F * (2.0 / B_m)));
	else
		tmp = (-2.0 / B_m) * sqrt((C * F));
	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[C, 2.55e+92], (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(-2.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 2.5500000000000001e92

   1. Initial program 20.2%

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

   3. Taylor expanded in 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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 16.0

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

   5. Applied rewrites 16.0%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 16.0

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 16.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 16.0

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

   7. Applied rewrites 16.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 16.0

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

   9. Applied rewrites 16.0%

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

   if 2.5500000000000001e92 < C

   1. Initial program 13.2%

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

   4. Applied rewrites 13.1%

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

   5. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 1.4

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

   7. Applied rewrites 1.4%

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

   8. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-*.f64 4.5

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

   10. Applied rewrites 4.5%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. distribute-lft-neg-in N/A

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

       8. lower-*.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. rem-square-sqrt N/A

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

       14. distribute-neg-frac N/A

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

       15. lower-/.f64 N/A

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

       16. metadata-eval 4.6

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

       17. lift-*.f64 N/A

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

       18. *-commutative N/A

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

       19. lower-*.f64 4.6

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

   12. Applied rewrites 4.6%

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

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B} \cdot \sqrt{C \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.4% 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 19.0%

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

3. Taylor expanded in 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

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 14.2

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

5. Applied rewrites 14.2%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 14.2

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

   6. lift-*.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. sqrt-unprod N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{F}{B}}}\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   15. lower-/.f64 14.2

       \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]

   16. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{2 \cdot F}}{B}}\right) \]

   18. lower-*.f64 14.2

       \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied rewrites 14.2%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   4. lower-/.f64 14.2

      \[\leadsto -\sqrt{F \cdot \color{blue}{\frac{2}{B}}} \]

9. Applied rewrites 14.2%

   \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))