ABCF->ab-angle a

Percentage Accurate: 19.0% → 52.4%

Time: 17.8s

Alternatives: 10

Speedup: 16.9×

• could not determine a ground truth

  1. A = -2.542687091886861e-184
  2. B = -1.2447986563389958e-57
  3. C = -3.546907088031563e+69
  4. F = 1.1796642671782842e-153

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 52.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (<= B_m 4.8e-7)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 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 (B_m <= 4.8e-7) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 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) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if (b_m <= 4.8d-7) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m ** 2.0d0))
    else
        tmp = -sqrt(f) / sqrt((b_m * 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 t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 4.8e-7) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = -Math.sqrt(F) / Math.sqrt((B_m * 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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 4.8e-7:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = -math.sqrt(F) / math.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 (B_m <= 4.8e-7)
		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)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 4.8e-7)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = -sqrt(F) / sqrt((B_m * 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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 4.8e-7], 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], 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 B < 4.79999999999999957e-7

   1. Initial program 20.5%

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

   3. Taylor expanded in 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 17.5

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

   5. Simplified 17.5%

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

   if 4.79999999999999957e-7 < B

   1. Initial program 16.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      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 43.6

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

   5. Simplified 43.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. unpow-prod-down N/A

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. metadata-eval N/A

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

      25. lower-sqrt.f64 57.8

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

   7. Applied egg-rr 57.8%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. sqrt-prod N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 58.0

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

   9. Applied egg-rr 58.0%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 58.0

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

   11. Applied egg-rr 58.0%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(0, A, 2 \cdot C\right)\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)} \cdot \sqrt{2}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (if (<= B_m 4.8e-7)
   (/
    (*
     (sqrt (* (* F (fma 0.0 A (* 2.0 C))) (fma -4.0 (* A C) (* B_m B_m))))
     (sqrt 2.0))
    (- (* (* 4.0 A) C) (pow B_m 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 tmp;
	if (B_m <= 4.8e-7) {
		tmp = (sqrt(((F * fma(0.0, A, (2.0 * C))) * fma(-4.0, (A * C), (B_m * B_m)))) * sqrt(2.0)) / (((4.0 * A) * C) - pow(B_m, 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)
	tmp = 0.0
	if (B_m <= 4.8e-7)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * fma(0.0, A, Float64(2.0 * C))) * fma(-4.0, Float64(A * C), Float64(B_m * B_m)))) * sqrt(2.0)) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 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_] := If[LessEqual[B$95$m, 4.8e-7], N[(N[(N[Sqrt[N[(N[(F * N[(0.0 * A + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $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 B < 4.79999999999999957e-7

   1. Initial program 20.5%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 11.4

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

   5. Simplified 11.4%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   8. Simplified 17.7%

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

   if 4.79999999999999957e-7 < B

   1. Initial program 16.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      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 43.6

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

   5. Simplified 43.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. unpow-prod-down N/A

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. metadata-eval N/A

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

      25. lower-sqrt.f64 57.8

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

   7. Applied egg-rr 57.8%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. sqrt-prod N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 58.0

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

   9. Applied egg-rr 58.0%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 58.0

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

   11. Applied egg-rr 58.0%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(0, A, 2 \cdot C\right)\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{2}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (if (<= (pow B_m 2.0) 2e-13)
   (* (sqrt (* F (/ (* (* B_m B_m) -0.5) A))) (- (/ (sqrt 2.0) B_m)))
   (/ (- (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 tmp;
	if (pow(B_m, 2.0) <= 2e-13) {
		tmp = sqrt((F * (((B_m * B_m) * -0.5) / A))) * -(sqrt(2.0) / B_m);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 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 ((b_m ** 2.0d0) <= 2d-13) then
        tmp = sqrt((f * (((b_m * b_m) * (-0.5d0)) / a))) * -(sqrt(2.0d0) / b_m)
    else
        tmp = -sqrt(f) / sqrt((b_m * 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 (Math.pow(B_m, 2.0) <= 2e-13) {
		tmp = Math.sqrt((F * (((B_m * B_m) * -0.5) / A))) * -(Math.sqrt(2.0) / B_m);
	} else {
		tmp = -Math.sqrt(F) / Math.sqrt((B_m * 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 math.pow(B_m, 2.0) <= 2e-13:
		tmp = math.sqrt((F * (((B_m * B_m) * -0.5) / A))) * -(math.sqrt(2.0) / B_m)
	else:
		tmp = -math.sqrt(F) / math.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)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-13)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(B_m * B_m) * -0.5) / A))) * Float64(-Float64(sqrt(2.0) / B_m)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 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 ((B_m ^ 2.0) <= 2e-13)
		tmp = sqrt((F * (((B_m * B_m) * -0.5) / A))) * -(sqrt(2.0) / B_m);
	else
		tmp = -sqrt(F) / sqrt((B_m * 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[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-13], N[(N[Sqrt[N[(F * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $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 (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-13

   1. Initial program 24.9%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 6.7

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

   5. Simplified 6.7%

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

   6. Taylor expanded in A around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 7.2

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}} \]

   8. Simplified 7.2%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5 \cdot \left(B \cdot B\right)}{A}}} \]

   if 2.0000000000000001e-13 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 19.2

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

   5. Simplified 19.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. unpow-prod-down N/A

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. metadata-eval N/A

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

      25. lower-sqrt.f64 24.2

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

   7. Applied egg-rr 24.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. sqrt-prod N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 24.2

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

   9. Applied egg-rr 24.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 24.2

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

   11. Applied egg-rr 24.2%

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

4. Final simplification 16.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(B \cdot B\right) \cdot -0.5}{A}} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;B\_m \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{t\_0 \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= B_m 4.8e-7)
     (* (/ (sqrt 2.0) -1.0) (/ (sqrt (* t_0 (* F (* 2.0 C)))) t_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 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (B_m <= 4.8e-7) {
		tmp = (sqrt(2.0) / -1.0) * (sqrt((t_0 * (F * (2.0 * C)))) / t_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 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B_m <= 4.8e-7)
		tmp = Float64(Float64(sqrt(2.0) / -1.0) * Float64(sqrt(Float64(t_0 * Float64(F * Float64(2.0 * C)))) / t_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[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.8e-7], N[(N[(N[Sqrt[2.0], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(t$95$0 * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$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 B < 4.79999999999999957e-7

   1. Initial program 20.5%

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

   4. Applied egg-rr 20.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 17.7

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

   7. Simplified 17.7%

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

   if 4.79999999999999957e-7 < B

   1. Initial program 16.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      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 43.6

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

   5. Simplified 43.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. unpow-prod-down N/A

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. metadata-eval N/A

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

      25. lower-sqrt.f64 57.8

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

   7. Applied egg-rr 57.8%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. sqrt-prod N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 58.0

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

   9. Applied egg-rr 58.0%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 58.0

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

   11. Applied egg-rr 58.0%

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

Alternative 5: 43.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.4 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\frac{F \cdot \mathsf{fma}\left(0, A, 2 \cdot C\right)}{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (if (<= B_m 5.4e-49)
   (*
    (sqrt (/ (* F (fma 0.0 A (* 2.0 C))) (fma -4.0 (* A C) (* B_m B_m))))
    (- (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 tmp;
	if (B_m <= 5.4e-49) {
		tmp = sqrt(((F * fma(0.0, A, (2.0 * C))) / fma(-4.0, (A * C), (B_m * B_m)))) * -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)
	tmp = 0.0
	if (B_m <= 5.4e-49)
		tmp = Float64(sqrt(Float64(Float64(F * fma(0.0, A, Float64(2.0 * C))) / fma(-4.0, Float64(A * C), Float64(B_m * B_m)))) * 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_] := If[LessEqual[B$95$m, 5.4e-49], N[(N[Sqrt[N[(N[(F * N[(0.0 * A + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $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 B < 5.3999999999999999e-49

   1. Initial program 19.8%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 11.2

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

   5. Simplified 11.2%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   8. Simplified 12.4%

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

   if 5.3999999999999999e-49 < B

   1. Initial program 19.5%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 41.7

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

   5. Simplified 41.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. unpow-prod-down N/A

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. pow1/2 N/A

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

      14. sqrt-unprod N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. sqrt-div N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. lower-/.f64 N/A

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

      24. metadata-eval N/A

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

      25. lower-sqrt.f64 54.1

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

   7. Applied egg-rr 54.1%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-div N/A

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

      4. sqrt-prod N/A

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

      5. div-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 54.3

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

   9. Applied egg-rr 54.3%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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. sqrt-div N/A

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

       9. metadata-eval N/A

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

       10. un-div-inv N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-sqrt.f64 N/A

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

       14. div-inv N/A

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

       15. metadata-eval N/A

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

       16. lower-*.f64 54.3

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

   11. Applied egg-rr 54.3%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.4 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\frac{F \cdot \mathsf{fma}\left(0, A, 2 \cdot C\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.9% 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 19.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   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 12.5

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

5. Simplified 12.5%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
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. pow1/2 N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. associate-*r* N/A

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

   7. unpow-prod-down N/A

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

   8. pow-prod-down N/A

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

   9. pow1/2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. pow1/2 N/A

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

   14. sqrt-unprod N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. pow1/2 N/A

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

   18. metadata-eval N/A

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

   19. sqrt-div N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. lower-/.f64 N/A

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

   24. metadata-eval N/A

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

   25. lower-sqrt.f64 15.8

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

7. Applied egg-rr 15.8%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-div N/A

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

   4. sqrt-prod N/A

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

   5. div-inv N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. sqrt-prod N/A

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

   10. pow1/2 N/A

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

   11. lower-*.f64 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-/.f64 15.9

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

9. Applied egg-rr 15.9%

   \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
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. sqrt-div N/A

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

    9. metadata-eval N/A

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

    10. un-div-inv N/A

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

    11. lower-/.f64 N/A

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

    12. lower-neg.f64 N/A

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

    13. lower-sqrt.f64 N/A

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

    14. div-inv N/A

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

    15. metadata-eval N/A

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

    16. lower-*.f64 15.9

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

11. Applied egg-rr 15.9%

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

Alternative 7: 35.9% accurate, 12.6× speedup

Math

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

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 19.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   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 12.5

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

5. Simplified 12.5%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
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. pow1/2 N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. associate-*r* N/A

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

   7. unpow-prod-down N/A

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

   8. pow-prod-down N/A

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

   9. pow1/2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. pow1/2 N/A

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

   14. sqrt-unprod N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. pow1/2 N/A

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

   18. metadata-eval N/A

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

   19. sqrt-div N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. lower-/.f64 N/A

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

   24. metadata-eval N/A

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

   25. lower-sqrt.f64 15.8

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

7. Applied egg-rr 15.8%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-div N/A

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

   4. sqrt-prod N/A

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

   5. div-inv N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. sqrt-prod N/A

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

   10. pow1/2 N/A

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

   11. lower-*.f64 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-/.f64 15.9

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

9. Applied egg-rr 15.9%

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

Alternative 8: 28.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.8 \cdot 10^{+239}:\\ \;\;\;\;-\sqrt{\left|\frac{2 \cdot F}{B\_m}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{-2}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 1.8e+239)
   (- (sqrt (fabs (/ (* 2.0 F) B_m))))
   (* (sqrt (* C 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) {
	double tmp;
	if (C <= 1.8e+239) {
		tmp = -sqrt(fabs(((2.0 * F) / B_m)));
	} else {
		tmp = sqrt((C * F)) * (-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 <= 1.8d+239) then
        tmp = -sqrt(abs(((2.0d0 * f) / b_m)))
    else
        tmp = sqrt((c * f)) * ((-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 <= 1.8e+239) {
		tmp = -Math.sqrt(Math.abs(((2.0 * F) / B_m)));
	} else {
		tmp = Math.sqrt((C * F)) * (-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 <= 1.8e+239:
		tmp = -math.sqrt(math.fabs(((2.0 * F) / B_m)))
	else:
		tmp = math.sqrt((C * F)) * (-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 <= 1.8e+239)
		tmp = Float64(-sqrt(abs(Float64(Float64(2.0 * F) / B_m))));
	else
		tmp = Float64(sqrt(Float64(C * F)) * 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 <= 1.8e+239)
		tmp = -sqrt(abs(((2.0 * F) / B_m)));
	else
		tmp = sqrt((C * F)) * (-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, 1.8e+239], (-N[Sqrt[N[Abs[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.8e239

   1. Initial program 21.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 13.2

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

   5. Simplified 13.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      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 13.2

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

      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. lower-*.f64 13.3

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

   7. Applied egg-rr 13.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 13.3

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

      3. rem-square-sqrt N/A

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

      4. sqrt-unprod N/A

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

      5. rem-sqrt-square N/A

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

      6. lower-fabs.f64 27.9

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 27.9

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

   9. Applied egg-rr 27.9%

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

   if 1.8e239 < C

   1. Initial program 1.4%

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

   3. Taylor expanded in C around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 12.8

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

   5. Simplified 12.8%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-+r+ N/A

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

      7. distribute-rgt1-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 1.6

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

   8. Simplified 1.6%

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

   9. Taylor expanded in B around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*r/ N/A

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

       8. unpow2 N/A

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

       9. rem-square-sqrt N/A

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

       10. metadata-eval N/A

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

       11. lower-/.f64 6.7

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

   11. Simplified 6.7%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.8 \cdot 10^{+239}:\\ \;\;\;\;-\sqrt{\left|\frac{2 \cdot F}{B}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{-2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.0% accurate, 15.8× speedup

Math

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

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 (fabs (/ (* 2.0 F) B_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   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 12.5

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

5. Simplified 12.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   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 12.5

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

   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. lower-*.f64 12.6

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

7. Applied egg-rr 12.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 12.6

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

   3. rem-square-sqrt N/A

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

   4. sqrt-unprod N/A

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

   5. rem-sqrt-square N/A

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

   6. lower-fabs.f64 26.6

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*r/ N/A

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

   10. lift-*.f64 N/A

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

   11. lower-/.f64 26.6

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

9. Applied egg-rr 26.6%

   \[\leadsto -\sqrt{\color{blue}{\left|\frac{2 \cdot F}{B}\right|}} \]
10. Add Preprocessing

Alternative 10: 27.8% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   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 12.5

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

5. Simplified 12.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   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 12.5

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

   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. lower-*.f64 12.6

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

7. Applied egg-rr 12.6%

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

Reproduce

herbie shell --seed 2024207 
(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))))