ABCF->ab-angle b

Percentage Accurate: 19.0% → 47.5%

Time: 17.6s

Alternatives: 13

Speedup: 5.8×

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: 47.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 5.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(A + A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)\right)}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 5.5e-21)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (+ A A)))
      (- t_0 (pow B_m 2.0)))
     (- 0.0 (/ (sqrt (* 2.0 (* F (- A (hypot A B_m))))) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 5.5e-21) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (A + A))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = 0.0 - (sqrt((2.0 * (F * (A - hypot(A, B_m))))) / B_m);
	}
	return tmp;
}

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 <= 5.5e-21) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (A + A))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = 0.0 - (Math.sqrt((2.0 * (F * (A - Math.hypot(A, B_m))))) / 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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 5.5e-21:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (A + A))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = 0.0 - (math.sqrt((2.0 * (F * (A - math.hypot(A, B_m))))) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 5.5e-21)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(A + A))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(0.0 - Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(A, B_m))))) / 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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 5.5e-21)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (A + A))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = 0.0 - (sqrt((2.0 * (F * (A - hypot(A, B_m))))) / 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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 5.5e-21], 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[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 5.49999999999999977e-21

   1. Initial program 26.3%

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

   3. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + 1 \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      4. +-lowering-+.f64 20.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \mathsf{+.f64}\left(A, A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

   5. Simplified 20.1%

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

   if 5.49999999999999977e-21 < B

   1. Initial program 18.4%

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

   3. Taylor expanded in C around 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

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right) \]

   5. Simplified 53.0%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 53.1%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right), B\right) \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      5. sqrt-unprod N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2}\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right), 2\right)\right)\right), B\right) \]

      10. accelerator-lowering-hypot.f64 53.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right), 2\right)\right)\right), B\right) \]

   9. Applied egg-rr 53.2%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{A \cdot \left(F \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)\right)}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.2e-21)
   (/
    (* -2.0 (sqrt (* A (* F (+ (* B_m B_m) (* -4.0 (* A C)))))))
    (- (* B_m B_m) (* (* 4.0 A) C)))
   (- 0.0 (/ (sqrt (* 2.0 (* F (- A (hypot A B_m))))) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.2e-21) {
		tmp = (-2.0 * sqrt((A * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / ((B_m * B_m) - ((4.0 * A) * C));
	} else {
		tmp = 0.0 - (sqrt((2.0 * (F * (A - hypot(A, B_m))))) / B_m);
	}
	return tmp;
}

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 (B_m <= 4.2e-21) {
		tmp = (-2.0 * Math.sqrt((A * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / ((B_m * B_m) - ((4.0 * A) * C));
	} else {
		tmp = 0.0 - (Math.sqrt((2.0 * (F * (A - Math.hypot(A, B_m))))) / 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 B_m <= 4.2e-21:
		tmp = (-2.0 * math.sqrt((A * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / ((B_m * B_m) - ((4.0 * A) * C))
	else:
		tmp = 0.0 - (math.sqrt((2.0 * (F * (A - math.hypot(A, B_m))))) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.2e-21)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(A * Float64(F * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))))) / Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(0.0 - Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(A, B_m))))) / 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 (B_m <= 4.2e-21)
		tmp = (-2.0 * sqrt((A * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / ((B_m * B_m) - ((4.0 * A) * C));
	else
		tmp = 0.0 - (sqrt((2.0 * (F * (A - hypot(A, B_m))))) / 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[B$95$m, 4.2e-21], N[(N[(-2.0 * N[Sqrt[N[(A * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.20000000000000025e-21

   1. Initial program 26.3%

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

   3. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + 1 \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      4. +-lowering-+.f64 20.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \mathsf{+.f64}\left(A, A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

   5. Simplified 20.1%

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

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}\right)}, \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(B, 2\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\left(A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, \color{blue}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left({B}^{2}\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left(B \cdot B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      11. *-lowering-*.f64 20.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

   8. Simplified 20.1%

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

      1. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\left(B \cdot B\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right)\right)\right) \]

      2. *-lowering-*.f64 20.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right)\right)\right) \]

   10. Applied egg-rr 20.1%

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

   if 4.20000000000000025e-21 < B

   1. Initial program 18.4%

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

   3. Taylor expanded in C around 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

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right) \]

   5. Simplified 53.0%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 53.1%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right), B\right) \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      5. sqrt-unprod N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2}\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right), 2\right)\right)\right), B\right) \]

      10. accelerator-lowering-hypot.f64 53.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right), 2\right)\right)\right), B\right) \]

   9. Applied egg-rr 53.2%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{A \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 430:\\ \;\;\;\;\frac{-2 \cdot \sqrt{A \cdot \left(F \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right) + 2 \cdot \left(A \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 430.0)
   (/
    (* -2.0 (sqrt (* A (* F (+ (* B_m B_m) (* -4.0 (* A C)))))))
    (- (* B_m B_m) (* (* 4.0 A) C)))
   (/ (sqrt (+ (* -2.0 (* B_m F)) (* 2.0 (* A F)))) (- 0.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 430.0) {
		tmp = (-2.0 * sqrt((A * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / ((B_m * B_m) - ((4.0 * A) * C));
	} else {
		tmp = sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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 (b_m <= 430.0d0) then
        tmp = ((-2.0d0) * sqrt((a * (f * ((b_m * b_m) + ((-4.0d0) * (a * c))))))) / ((b_m * b_m) - ((4.0d0 * a) * c))
    else
        tmp = sqrt((((-2.0d0) * (b_m * f)) + (2.0d0 * (a * f)))) / (0.0d0 - 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 (B_m <= 430.0) {
		tmp = (-2.0 * Math.sqrt((A * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / ((B_m * B_m) - ((4.0 * A) * C));
	} else {
		tmp = Math.sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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 B_m <= 430.0:
		tmp = (-2.0 * math.sqrt((A * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / ((B_m * B_m) - ((4.0 * A) * C))
	else:
		tmp = math.sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 430.0)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(A * Float64(F * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))))) / Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(sqrt(Float64(Float64(-2.0 * Float64(B_m * F)) + Float64(2.0 * Float64(A * F)))) / Float64(0.0 - 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 (B_m <= 430.0)
		tmp = (-2.0 * sqrt((A * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / ((B_m * B_m) - ((4.0 * A) * C));
	else
		tmp = sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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[B$95$m, 430.0], N[(N[(-2.0 * N[Sqrt[N[(A * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 430

   1. Initial program 27.0%

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

   3. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + 1 \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      4. +-lowering-+.f64 20.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \mathsf{+.f64}\left(A, A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

   5. Simplified 20.0%

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

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}\right)}, \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(B, 2\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\left(A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, \color{blue}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left({B}^{2}\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left(B \cdot B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      11. *-lowering-*.f64 20.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

   8. Simplified 20.0%

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

      1. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\left(B \cdot B\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right)\right)\right) \]

      2. *-lowering-*.f64 20.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right)\right)\right) \]

   10. Applied egg-rr 20.0%

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

   if 430 < B

   1. Initial program 15.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right) \]

   5. Simplified 54.7%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 54.7%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right), B\right) \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      5. sqrt-unprod N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2}\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right), 2\right)\right)\right), B\right) \]

      10. accelerator-lowering-hypot.f64 54.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right), 2\right)\right)\right), B\right) \]

   9. Applied egg-rr 54.8%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right) + 2 \cdot \left(A \cdot F\right)\right)}\right)\right), B\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot \left(B \cdot F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

       5. *-lowering-*.f64 51.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right), B\right) \]

   12. Simplified 51.3%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 430:\\ \;\;\;\;\frac{-2 \cdot \sqrt{A \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right) + 2 \cdot \left(A \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 380:\\ \;\;\;\;\sqrt{t\_0 \cdot \left(A \cdot F\right)} \cdot \frac{-2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right) + 2 \cdot \left(A \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (* B_m B_m) (* -4.0 (* A C)))))
   (if (<= B_m 380.0)
     (* (sqrt (* t_0 (* A F))) (/ -2.0 t_0))
     (/ (sqrt (+ (* -2.0 (* B_m F)) (* 2.0 (* A F)))) (- 0.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 380.0) {
		tmp = sqrt((t_0 * (A * F))) * (-2.0 / t_0);
	} else {
		tmp = sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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) :: t_0
    real(8) :: tmp
    t_0 = (b_m * b_m) + ((-4.0d0) * (a * c))
    if (b_m <= 380.0d0) then
        tmp = sqrt((t_0 * (a * f))) * ((-2.0d0) / t_0)
    else
        tmp = sqrt((((-2.0d0) * (b_m * f)) + (2.0d0 * (a * f)))) / (0.0d0 - 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 t_0 = (B_m * B_m) + (-4.0 * (A * C));
	double tmp;
	if (B_m <= 380.0) {
		tmp = Math.sqrt((t_0 * (A * F))) * (-2.0 / t_0);
	} else {
		tmp = Math.sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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):
	t_0 = (B_m * B_m) + (-4.0 * (A * C))
	tmp = 0
	if B_m <= 380.0:
		tmp = math.sqrt((t_0 * (A * F))) * (-2.0 / t_0)
	else:
		tmp = math.sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B_m <= 380.0)
		tmp = Float64(sqrt(Float64(t_0 * Float64(A * F))) * Float64(-2.0 / t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(-2.0 * Float64(B_m * F)) + Float64(2.0 * Float64(A * F)))) / Float64(0.0 - 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)
	t_0 = (B_m * B_m) + (-4.0 * (A * C));
	tmp = 0.0;
	if (B_m <= 380.0)
		tmp = sqrt((t_0 * (A * F))) * (-2.0 / t_0);
	else
		tmp = sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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_] := Block[{t$95$0 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 380.0], N[(N[Sqrt[N[(t$95$0 * N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 380

   1. Initial program 27.0%

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

   3. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + 1 \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      4. +-lowering-+.f64 20.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \mathsf{+.f64}\left(A, A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

   5. Simplified 20.0%

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

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}\right)}, \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{pow.f64}\left(B, 2\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\left(A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, \color{blue}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left({B}^{2}\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left(B \cdot B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      11. *-lowering-*.f64 20.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

   8. Simplified 20.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{A \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}\right), \color{blue}{\left(\frac{-2}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)}\right) \]

   10. Applied egg-rr 18.5%

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

   if 380 < B

   1. Initial program 15.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right) \]

   5. Simplified 54.7%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 54.7%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right), B\right) \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      5. sqrt-unprod N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2}\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right), 2\right)\right)\right), B\right) \]

      10. accelerator-lowering-hypot.f64 54.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right), 2\right)\right)\right), B\right) \]

   9. Applied egg-rr 54.8%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right) + 2 \cdot \left(A \cdot F\right)\right)}\right)\right), B\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot \left(B \cdot F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

       5. *-lowering-*.f64 51.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right), B\right) \]

   12. Simplified 51.3%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 380:\\ \;\;\;\;\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A \cdot F\right)} \cdot \frac{-2}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right) + 2 \cdot \left(A \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 370:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right) + 2 \cdot \left(A \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 370.0)
   (* -2.0 (sqrt (/ (* A F) (+ (* B_m B_m) (* -4.0 (* A C))))))
   (/ (sqrt (+ (* -2.0 (* B_m F)) (* 2.0 (* A F)))) (- 0.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 370.0) {
		tmp = -2.0 * sqrt(((A * F) / ((B_m * B_m) + (-4.0 * (A * C)))));
	} else {
		tmp = sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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 (b_m <= 370.0d0) then
        tmp = (-2.0d0) * sqrt(((a * f) / ((b_m * b_m) + ((-4.0d0) * (a * c)))))
    else
        tmp = sqrt((((-2.0d0) * (b_m * f)) + (2.0d0 * (a * f)))) / (0.0d0 - 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 (B_m <= 370.0) {
		tmp = -2.0 * Math.sqrt(((A * F) / ((B_m * B_m) + (-4.0 * (A * C)))));
	} else {
		tmp = Math.sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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 B_m <= 370.0:
		tmp = -2.0 * math.sqrt(((A * F) / ((B_m * B_m) + (-4.0 * (A * C)))))
	else:
		tmp = math.sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 370.0)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))));
	else
		tmp = Float64(sqrt(Float64(Float64(-2.0 * Float64(B_m * F)) + Float64(2.0 * Float64(A * F)))) / Float64(0.0 - 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 (B_m <= 370.0)
		tmp = -2.0 * sqrt(((A * F) / ((B_m * B_m) + (-4.0 * (A * C)))));
	else
		tmp = sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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[B$95$m, 370.0], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 370

   1. Initial program 27.0%

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

   3. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + 1 \cdot A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \left(A + A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

      4. +-lowering-+.f64 20.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right), F\right)\right), \mathsf{+.f64}\left(A, A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(B, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)\right)\right) \]

   5. Simplified 20.0%

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

   6. Taylor expanded in F around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(-2, \color{blue}{\left(\sqrt{\frac{A \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\left(\frac{A \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(A \cdot F\right), \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(A, F\right), \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(A, F\right), \left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(A, F\right), \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(A, F\right), \mathsf{+.f64}\left(\left({B}^{2}\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(A, F\right), \mathsf{+.f64}\left(\left(B \cdot B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(A, F\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(A, F\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 13.5%

          \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(A, F\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right)\right)\right)\right) \]

   8. Simplified 13.5%

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

   if 370 < B

   1. Initial program 15.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right) \]

   5. Simplified 54.7%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 54.7%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right), B\right) \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

      5. sqrt-unprod N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2}\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right), 2\right)\right)\right), B\right) \]

      10. accelerator-lowering-hypot.f64 54.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right), 2\right)\right)\right), B\right) \]

   9. Applied egg-rr 54.8%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right) + 2 \cdot \left(A \cdot F\right)\right)}\right)\right), B\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot \left(B \cdot F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

       5. *-lowering-*.f64 51.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right), B\right) \]

   12. Simplified 51.3%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 370:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{B \cdot B + -4 \cdot \left(A \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right) + 2 \cdot \left(A \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right) + 2 \cdot \left(A \cdot F\right)}}{0 - B\_m} \end{array} \]

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 (* B_m F)) (* 2.0 (* A F)))) (- 0.0 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(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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((((-2.0d0) * (b_m * f)) + (2.0d0 * (a * f)))) / (0.0d0 - 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(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(Float64(-2.0 * Float64(B_m * F)) + Float64(2.0 * Float64(A * F)))) / Float64(0.0 - 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(((-2.0 * (B_m * F)) + (2.0 * (A * F)))) / (0.0 - 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[N[(N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in C around 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

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right) \]

5. Simplified 16.9%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right), \color{blue}{B}\right) \]

7. Applied egg-rr 16.9%

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

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right), B\right) \]

   3. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

   5. sqrt-unprod N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2}\right)\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2\right)\right)\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right), 2\right)\right)\right), B\right) \]

   10. accelerator-lowering-hypot.f64 17.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right), 2\right)\right)\right), B\right) \]

9. Applied egg-rr 17.0%

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

10. Taylor expanded in A around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right) + 2 \cdot \left(A \cdot F\right)\right)}\right)\right), B\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot \left(B \cdot F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \left(2 \cdot \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \mathsf{*.f64}\left(2, \left(A \cdot F\right)\right)\right)\right)\right), B\right) \]

    5. *-lowering-*.f64 14.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(A, F\right)\right)\right)\right)\right), B\right) \]

12. Simplified 14.5%

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

13. Final simplification 14.5%

    \[\leadsto \frac{\sqrt{-2 \cdot \left(B \cdot F\right) + 2 \cdot \left(A \cdot F\right)}}{0 - B} \]
14. Add Preprocessing

Alternative 7: 26.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m} \end{array} \]

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 (* B_m F))) (- 0.0 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((-2.0 * (B_m * F))) / (0.0 - 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(((-2.0d0) * (b_m * f))) / (0.0d0 - 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((-2.0 * (B_m * F))) / (0.0 - 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((-2.0 * (B_m * F))) / (0.0 - B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - 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((-2.0 * (B_m * F))) / (0.0 - 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[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in C around 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

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right) \]

5. Simplified 16.9%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right), \color{blue}{B}\right) \]

7. Applied egg-rr 16.9%

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

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right), B\right) \]

   3. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)} \cdot \sqrt{2}\right)\right), B\right) \]

   5. sqrt-unprod N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2}\right)\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot 2\right)\right)\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{A \cdot A + B \cdot B}\right)\right), 2\right)\right)\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right), 2\right)\right)\right), B\right) \]

   10. accelerator-lowering-hypot.f64 17.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right), 2\right)\right)\right), B\right) \]

9. Applied egg-rr 17.0%

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

10. Taylor expanded in A around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right)\right), B\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right)\right), B\right) \]

    2. *-lowering-*.f64 15.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right), B\right) \]

12. Simplified 15.4%

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

13. Final simplification 15.4%

    \[\leadsto \frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B} \]
14. Add Preprocessing

Alternative 8: 8.7% accurate, 5.9× speedup

Math

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

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 (/ (* -2.0 (sqrt (* A F))) B_m))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return (-2.0 * sqrt((A * 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 = ((-2.0d0) * sqrt((a * 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 (-2.0 * Math.sqrt((A * 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 (-2.0 * math.sqrt((A * 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(Float64(-2.0 * sqrt(Float64(A * 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 = (-2.0 * sqrt((A * 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[(N[(-2.0 * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in C around 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

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right) \]

5. Simplified 16.9%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right), \color{blue}{B}\right) \]

7. Applied egg-rr 16.9%

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

8. Taylor expanded in A around -inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\sqrt{A \cdot F} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}\right)\right)}, B\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{A \cdot F} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {\left(\sqrt{2}\right)}^{2}\right)\right), B\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{A \cdot F} \cdot \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right)\right), B\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{A \cdot F} \cdot \left(-1 \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right)\right), B\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{A \cdot F} \cdot \left(-1 \cdot 2\right)\right), B\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{A \cdot F} \cdot -2\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{A \cdot F}\right), -2\right), B\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A \cdot F\right)\right), -2\right), B\right) \]

   8. *-lowering-*.f64 3.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, F\right)\right), -2\right), B\right) \]

10. Simplified 3.0%

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

11. Final simplification 3.0%

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

Alternative 9: 8.7% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in C around 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

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

   2. *-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + {A}^{2}}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), \mathsf{*.f64}\left(-1, \color{blue}{\left(\frac{\sqrt{2}}{B}\right)}\right)\right) \]

5. Simplified 16.9%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right), \color{blue}{B}\right) \]

7. Applied egg-rr 16.9%

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

8. Taylor expanded in A around -inf

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. rem-square-sqrt N/A

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

   5. metadata-eval N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{A \cdot F}\right), \color{blue}{\left(\frac{-2}{B}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A \cdot F\right)\right), \left(\frac{\color{blue}{-2}}{B}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, F\right)\right), \left(\frac{-2}{B}\right)\right) \]

   9. /-lowering-/.f64 3.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, F\right)\right), \mathsf{/.f64}\left(-2, \color{blue}{B}\right)\right) \]

10. Simplified 3.0%

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

Alternative 10: 1.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ {\left(2 \cdot \frac{F}{B\_m}\right)}^{0.5} \end{array} \]

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 (pow (* 2.0 (/ F 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 pow((2.0 * (F / 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 = (2.0d0 * (f / 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.pow((2.0 * (F / 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.pow((2.0 * (F / 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(2.0 * Float64(F / 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 = (2.0 * (F / 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[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \color{blue}{\left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{2}\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left(-1 \cdot \sqrt{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(-1, \left(\sqrt{2}\right)\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 1.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 1.7%

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

   1. pow1/2 N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. pow1/2 N/A

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

   5. pow-prod-down N/A

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

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

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{F}{B} \cdot 2\right), \color{blue}{\frac{1}{2}}\right) \]

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

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{B}\right), 2\right), \frac{1}{2}\right) \]

   8. /-lowering-/.f64 1.9%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), 2\right), \frac{1}{2}\right) \]

7. Applied egg-rr 1.9%

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

8. Final simplification 1.9%

   \[\leadsto {\left(2 \cdot \frac{F}{B}\right)}^{0.5} \]
9. Add Preprocessing

Alternative 11: 1.6% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \color{blue}{\left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{2}\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left(-1 \cdot \sqrt{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(-1, \left(\sqrt{2}\right)\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 1.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 1.7%

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

   1. mul-1-neg N/A

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

   2. remove-double-neg N/A

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

   3. sqrt-unprod N/A

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

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

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{F}{B} \cdot 2\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{B}\right), 2\right)\right) \]

   6. /-lowering-/.f64 1.7%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), 2\right)\right) \]

7. Applied egg-rr 1.7%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{F}{B}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{1}{\frac{B}{F}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{B}{F}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{B}{F}\right)\right)\right) \]

   5. /-lowering-/.f64 1.8%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(B, F\right)\right)\right) \]

9. Applied egg-rr 1.8%

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

Alternative 12: 1.6% accurate, 6.0× 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 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 24.4%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \color{blue}{\left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{2}\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left(-1 \cdot \sqrt{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(-1, \left(\sqrt{2}\right)\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 1.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 1.7%

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

   1. mul-1-neg N/A

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

   2. remove-double-neg N/A

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

   3. sqrt-unprod N/A

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

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

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{F}{B} \cdot 2\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{B}\right), 2\right)\right) \]

   6. /-lowering-/.f64 1.7%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), 2\right)\right) \]

7. Applied egg-rr 1.7%

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

8. Final simplification 1.7%

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

Alternative 13: 1.6% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \color{blue}{\left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{2}\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\left(-1 \cdot \sqrt{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(-1, \left(\sqrt{2}\right)\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 1.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 1.7%

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

   1. mul-1-neg N/A

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

   2. remove-double-neg N/A

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

   3. sqrt-unprod N/A

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

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

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{F}{B} \cdot 2\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{B}\right), 2\right)\right) \]

   6. /-lowering-/.f64 1.7%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), 2\right)\right) \]

7. Applied egg-rr 1.7%

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

   1. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{F \cdot 2}{B}\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(F \cdot \frac{2}{B}\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{2}{B}\right)\right)\right) \]

   4. /-lowering-/.f64 1.7%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(2, B\right)\right)\right) \]

9. Applied egg-rr 1.7%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))