ABCF->ab-angle b

Percentage Accurate: 18.6% → 50.2%

Time: 23.1s

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: 18.6% 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: 50.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := t\_2 - B\_m \cdot B\_m\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \left(t\_1 - C\right)\right)\right)}}{t\_4}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - t\_1\right)}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (* B_m B_m) (* C (* A -4.0))))
        (t_1 (hypot B_m (- A C)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0))))
        (t_4 (- t_2 (* B_m B_m))))
   (if (<= t_3 0.0)
     (/ (* (sqrt t_0) (sqrt (* 2.0 (* F (- A (- t_1 C)))))) t_4)
     (if (<= t_3 INFINITY)
       (/ (sqrt (* (* t_0 (* 2.0 F)) (- (+ A C) t_1))) t_4)
       (/ (* (sqrt (* F -2.0)) (sqrt B_m)) (- 0.0 B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) + (C * (A * -4.0));
	double t_1 = hypot(B_m, (A - C));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double t_4 = t_2 - (B_m * B_m);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = (sqrt(t_0) * sqrt((2.0 * (F * (A - (t_1 - C)))))) / t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) - t_1))) / t_4;
	} else {
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) + (C * (A * -4.0));
	double t_1 = Math.hypot(B_m, (A - C));
	double t_2 = (4.0 * A) * C;
	double t_3 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_2) * F)) * ((A + C) - Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_2 - Math.pow(B_m, 2.0));
	double t_4 = t_2 - (B_m * B_m);
	double tmp;
	if (t_3 <= 0.0) {
		tmp = (Math.sqrt(t_0) * Math.sqrt((2.0 * (F * (A - (t_1 - C)))))) / t_4;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((t_0 * (2.0 * F)) * ((A + C) - t_1))) / t_4;
	} else {
		tmp = (Math.sqrt((F * -2.0)) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (B_m * B_m) + (C * (A * -4.0))
	t_1 = math.hypot(B_m, (A - C))
	t_2 = (4.0 * A) * C
	t_3 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_2) * F)) * ((A + C) - math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_2 - math.pow(B_m, 2.0))
	t_4 = t_2 - (B_m * B_m)
	tmp = 0
	if t_3 <= 0.0:
		tmp = (math.sqrt(t_0) * math.sqrt((2.0 * (F * (A - (t_1 - C)))))) / t_4
	elif t_3 <= math.inf:
		tmp = math.sqrt(((t_0 * (2.0 * F)) * ((A + C) - t_1))) / t_4
	else:
		tmp = (math.sqrt((F * -2.0)) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

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

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m * B_m) + (C * (A * -4.0));
	t_1 = hypot(B_m, (A - C));
	t_2 = (4.0 * A) * C;
	t_3 = sqrt(((2.0 * (((B_m ^ 2.0) - t_2) * F)) * ((A + C) - sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_2 - (B_m ^ 2.0));
	t_4 = t_2 - (B_m * B_m);
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = (sqrt(t_0) * sqrt((2.0 * (F * (A - (t_1 - C)))))) / t_4;
	elseif (t_3 <= Inf)
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) - t_1))) / t_4;
	else
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * N[(A - N[(t$95$1 - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 0.0

   1. Initial program 34.8%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 38.5%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. unpow-prod-down N/A

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

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

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

   6. Applied egg-rr 51.8%

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

   if 0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 46.2%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 57.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 14.8%

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

   5. Simplified 14.8%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 14.9%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 13.9%

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

   10. Simplified 13.9%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
   11. Step-by-step derivation

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. unpow1/2 N/A

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

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

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

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

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

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

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

       9. unpow1/2 N/A

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

       10. sqrt-lowering-sqrt.f64 21.6%

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

   12. Applied egg-rr 21.6%

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

4. Final simplification 39.6%

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

Alternative 2: 44.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 0.1:\\ \;\;\;\;\frac{-1}{\frac{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)}{\sqrt{2 \cdot \left(\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 0.1)
   (/
    -1.0
    (/
     (- (* B_m B_m) (* 4.0 (* A C)))
     (sqrt
      (*
       2.0
       (*
        (+ (* B_m B_m) (* C (* A -4.0)))
        (* F (- A (- (hypot B_m (- A C)) C))))))))
   (/ (* (sqrt (* F -2.0)) (sqrt B_m)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 0.1) {
		tmp = -1.0 / (((B_m * B_m) - (4.0 * (A * C))) / sqrt((2.0 * (((B_m * B_m) + (C * (A * -4.0))) * (F * (A - (hypot(B_m, (A - C)) - C)))))));
	} else {
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 0.1) {
		tmp = -1.0 / (((B_m * B_m) - (4.0 * (A * C))) / Math.sqrt((2.0 * (((B_m * B_m) + (C * (A * -4.0))) * (F * (A - (Math.hypot(B_m, (A - C)) - C)))))));
	} else {
		tmp = (Math.sqrt((F * -2.0)) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 0.1:
		tmp = -1.0 / (((B_m * B_m) - (4.0 * (A * C))) / math.sqrt((2.0 * (((B_m * B_m) + (C * (A * -4.0))) * (F * (A - (math.hypot(B_m, (A - C)) - C)))))))
	else:
		tmp = (math.sqrt((F * -2.0)) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 0.1)
		tmp = Float64(-1.0 / Float64(Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C))) / sqrt(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(F * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C))))))));
	else
		tmp = Float64(Float64(sqrt(Float64(F * -2.0)) * sqrt(B_m)) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 0.1)
		tmp = -1.0 / (((B_m * B_m) - (4.0 * (A * C))) / sqrt((2.0 * (((B_m * B_m) + (C * (A * -4.0))) * (F * (A - (hypot(B_m, (A - C)) - C)))))));
	else
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.1], N[(-1.0 / N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 0.10000000000000001

   1. Initial program 26.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 31.3%

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

   6. Applied egg-rr 31.9%

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

   if 0.10000000000000001 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.9%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 22.5%

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

   5. Simplified 22.5%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 22.6%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 21.5%

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

   10. Simplified 21.5%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
   11. Step-by-step derivation

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. unpow1/2 N/A

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

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

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

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

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

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

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

       9. unpow1/2 N/A

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

       10. sqrt-lowering-sqrt.f64 30.9%

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

   12. Applied egg-rr 30.9%

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

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 0.1:\\ \;\;\;\;\frac{-1}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\sqrt{2 \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4500000000:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)\right)\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4500000000.0)
   (*
    (sqrt
     (*
      2.0
      (*
       (+ (* B_m B_m) (* C (* A -4.0)))
       (* F (- A (- (hypot B_m (- A C)) C))))))
    (/ 1.0 (- (* 4.0 (* A C)) (* B_m B_m))))
   (/ (* (sqrt (* F -2.0)) (sqrt B_m)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4500000000.0) {
		tmp = sqrt((2.0 * (((B_m * B_m) + (C * (A * -4.0))) * (F * (A - (hypot(B_m, (A - C)) - C)))))) * (1.0 / ((4.0 * (A * C)) - (B_m * B_m)));
	} else {
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4500000000.0) {
		tmp = Math.sqrt((2.0 * (((B_m * B_m) + (C * (A * -4.0))) * (F * (A - (Math.hypot(B_m, (A - C)) - C)))))) * (1.0 / ((4.0 * (A * C)) - (B_m * B_m)));
	} else {
		tmp = (Math.sqrt((F * -2.0)) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4500000000.0:
		tmp = math.sqrt((2.0 * (((B_m * B_m) + (C * (A * -4.0))) * (F * (A - (math.hypot(B_m, (A - C)) - C)))))) * (1.0 / ((4.0 * (A * C)) - (B_m * B_m)))
	else:
		tmp = (math.sqrt((F * -2.0)) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4500000000.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(F * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C)))))) * Float64(1.0 / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(F * -2.0)) * sqrt(B_m)) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4500000000.0)
		tmp = sqrt((2.0 * (((B_m * B_m) + (C * (A * -4.0))) * (F * (A - (hypot(B_m, (A - C)) - C)))))) * (1.0 / ((4.0 * (A * C)) - (B_m * B_m)));
	else
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4500000000.0], N[(N[Sqrt[N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.5e9

   1. Initial program 22.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 25.8%

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

   6. Applied egg-rr 26.3%

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

   if 4.5e9 < B

   1. Initial program 17.6%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 46.6%

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

   5. Simplified 46.6%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 46.9%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 47.2%

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

   10. Simplified 47.2%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
   11. Step-by-step derivation

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. unpow1/2 N/A

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

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

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

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

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

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

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

       9. unpow1/2 N/A

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

       10. sqrt-lowering-sqrt.f64 68.7%

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

   12. Applied egg-rr 68.7%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4500000000:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 13200000:\\ \;\;\;\;\frac{\sqrt{\left(\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 13200000.0)
   (/
    (sqrt
     (*
      (* (+ (* B_m B_m) (* C (* A -4.0))) (* 2.0 F))
      (- (+ A C) (hypot B_m (- A C)))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (* (sqrt (* F -2.0)) (sqrt B_m)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 13200000.0) {
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * F)) * ((A + C) - hypot(B_m, (A - C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 13200000.0) {
		tmp = Math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * F)) * ((A + C) - Math.hypot(B_m, (A - C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (Math.sqrt((F * -2.0)) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 13200000.0:
		tmp = math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * F)) * ((A + C) - math.hypot(B_m, (A - C))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = (math.sqrt((F * -2.0)) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 13200000.0)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(2.0 * F)) * Float64(Float64(A + C) - hypot(B_m, Float64(A - C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(F * -2.0)) * sqrt(B_m)) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 13200000.0)
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * F)) * ((A + C) - hypot(B_m, (A - C))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 13200000.0], N[(N[Sqrt[N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.32e7

   1. Initial program 22.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 25.8%

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

   if 1.32e7 < B

   1. Initial program 17.6%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 46.6%

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

   5. Simplified 46.6%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 46.9%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 47.2%

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

   10. Simplified 47.2%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
   11. Step-by-step derivation

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. unpow1/2 N/A

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

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

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

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

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

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

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

       9. unpow1/2 N/A

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

       10. sqrt-lowering-sqrt.f64 68.7%

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

   12. Applied egg-rr 68.7%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 13200000:\\ \;\;\;\;\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{{\left(F \cdot \left(A \cdot C\right)\right)}^{0.5} \cdot \sqrt{C \cdot -16}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.4e-89)
   (/
    (* (pow (* F (* A C)) 0.5) (sqrt (* C -16.0)))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (* (sqrt (* F -2.0)) (sqrt B_m)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.4e-89) {
		tmp = (pow((F * (A * C)), 0.5) * sqrt((C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
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 <= 5.4d-89) then
        tmp = (((f * (a * c)) ** 0.5d0) * sqrt((c * (-16.0d0)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = (sqrt((f * (-2.0d0))) * sqrt(b_m)) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.4e-89) {
		tmp = (Math.pow((F * (A * C)), 0.5) * Math.sqrt((C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (Math.sqrt((F * -2.0)) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5.4e-89:
		tmp = (math.pow((F * (A * C)), 0.5) * math.sqrt((C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = (math.sqrt((F * -2.0)) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.4e-89)
		tmp = Float64(Float64((Float64(F * Float64(A * C)) ^ 0.5) * sqrt(Float64(C * -16.0))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(F * -2.0)) * sqrt(B_m)) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 5.4e-89)
		tmp = (((F * (A * C)) ^ 0.5) * sqrt((C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.4e-89], N[(N[(N[Power[N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(C * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.39999999999999975e-89

   1. Initial program 22.2%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 25.6%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.6%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(F \cdot A\right) \cdot \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot F\right) \cdot \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot F\right), \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot A\right), \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 10.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \mathsf{*.f64}\left(C, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 10.6%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left(F \cdot A\right) \cdot \left(C \cdot C\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(F \cdot A\right) \cdot \left(C \cdot C\right)\right) \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(\left(F \cdot A\right) \cdot C\right) \cdot C\right) \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(F \cdot A\right) \cdot C\right) \cdot \left(C \cdot -16\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. sqrt-prod N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(F \cdot A\right) \cdot C} \cdot \sqrt{C \cdot -16}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(F \cdot A\right) \cdot C\right)}^{\frac{1}{2}} \cdot \sqrt{C \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(\left(F \cdot A\right) \cdot C\right)}^{\frac{1}{2}}\right), \left(\sqrt{C \cdot -16}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(F \cdot A\right) \cdot C\right), \frac{1}{2}\right), \left(\sqrt{C \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(F \cdot \left(A \cdot C\right)\right), \frac{1}{2}\right), \left(\sqrt{C \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(A \cdot C\right)\right), \frac{1}{2}\right), \left(\sqrt{C \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(C \cdot A\right)\right), \frac{1}{2}\right), \left(\sqrt{C \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, A\right)\right), \frac{1}{2}\right), \left(\sqrt{C \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, A\right)\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       13. *-lowering-*.f64 14.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, A\right)\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(C, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 14.7%

       \[\leadsto \frac{\color{blue}{{\left(F \cdot \left(C \cdot A\right)\right)}^{0.5} \cdot \sqrt{C \cdot -16}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 5.39999999999999975e-89 < B

   1. Initial program 18.7%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 38.0%

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

   5. Simplified 38.0%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 38.2%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 38.6%

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

   10. Simplified 38.6%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
   11. Step-by-step derivation

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. unpow1/2 N/A

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

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

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

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

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

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

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

       9. unpow1/2 N/A

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

       10. sqrt-lowering-sqrt.f64 54.3%

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

   12. Applied egg-rr 54.3%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.4 \cdot 10^{-89}:\\ \;\;\;\;\frac{{\left(F \cdot \left(A \cdot C\right)\right)}^{0.5} \cdot \sqrt{C \cdot -16}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 10^{-134}:\\ \;\;\;\;\frac{{\left(C \cdot \left(A \cdot C\right)\right)}^{0.5} \cdot \sqrt{F \cdot -16}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1e-134)
   (/
    (* (pow (* C (* A C)) 0.5) (sqrt (* F -16.0)))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (* (sqrt (* F -2.0)) (sqrt B_m)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1e-134) {
		tmp = (pow((C * (A * C)), 0.5) * sqrt((F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
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 <= 1d-134) then
        tmp = (((c * (a * c)) ** 0.5d0) * sqrt((f * (-16.0d0)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = (sqrt((f * (-2.0d0))) * sqrt(b_m)) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1e-134) {
		tmp = (Math.pow((C * (A * C)), 0.5) * Math.sqrt((F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (Math.sqrt((F * -2.0)) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1e-134:
		tmp = (math.pow((C * (A * C)), 0.5) * math.sqrt((F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = (math.sqrt((F * -2.0)) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1e-134)
		tmp = Float64(Float64((Float64(C * Float64(A * C)) ^ 0.5) * sqrt(Float64(F * -16.0))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(F * -2.0)) * sqrt(B_m)) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1e-134)
		tmp = (((C * (A * C)) ^ 0.5) * sqrt((F * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1e-134], N[(N[(N[Power[N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(F * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.00000000000000004e-134

   1. Initial program 22.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 25.0%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 11.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.0%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right) \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -16\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{A \cdot \left(C \cdot C\right)} \cdot \sqrt{F \cdot -16}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(A \cdot \left(C \cdot C\right)\right)}^{\frac{1}{2}} \cdot \sqrt{F \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(A \cdot \left(C \cdot C\right)\right)}^{\frac{1}{2}}\right), \left(\sqrt{F \cdot -16}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(A \cdot \left(C \cdot C\right)\right), \frac{1}{2}\right), \left(\sqrt{F \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(A \cdot C\right) \cdot C\right), \frac{1}{2}\right), \left(\sqrt{F \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(A \cdot C\right)\right), \frac{1}{2}\right), \left(\sqrt{F \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(A \cdot C\right)\right), \frac{1}{2}\right), \left(\sqrt{F \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(C \cdot A\right)\right), \frac{1}{2}\right), \left(\sqrt{F \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right), \frac{1}{2}\right), \left(\sqrt{F \cdot -16}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. *-lowering-*.f64 12.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 12.6%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(C \cdot A\right)\right)}^{0.5} \cdot \sqrt{F \cdot -16}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.00000000000000004e-134 < B

   1. Initial program 19.5%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 36.7%

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

   5. Simplified 36.7%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 36.8%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 36.7%

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

   10. Simplified 36.7%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
   11. Step-by-step derivation

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. unpow1/2 N/A

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

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

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

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

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

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

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

       9. unpow1/2 N/A

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

       10. sqrt-lowering-sqrt.f64 52.0%

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

   12. Applied egg-rr 52.0%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 10^{-134}:\\ \;\;\;\;\frac{{\left(C \cdot \left(A \cdot C\right)\right)}^{0.5} \cdot \sqrt{F \cdot -16}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.3 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B\_m}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.3e-112)
   (/ (sqrt (* (* F (* A C)) (* C -16.0))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (* (sqrt (* F -2.0)) (sqrt B_m)) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.3e-112) {
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 2.3d-112) then
        tmp = sqrt(((f * (a * c)) * (c * (-16.0d0)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = (sqrt((f * (-2.0d0))) * sqrt(b_m)) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.3e-112) {
		tmp = Math.sqrt(((F * (A * C)) * (C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (Math.sqrt((F * -2.0)) * Math.sqrt(B_m)) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2.3e-112:
		tmp = math.sqrt(((F * (A * C)) * (C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = (math.sqrt((F * -2.0)) * math.sqrt(B_m)) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.3e-112)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A * C)) * Float64(C * -16.0))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(F * -2.0)) * sqrt(B_m)) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.3e-112)
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = (sqrt((F * -2.0)) * sqrt(B_m)) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.3e-112], N[(N[Sqrt[N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(C * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.29999999999999991e-112

   1. Initial program 21.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 25.0%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(F \cdot A\right) \cdot \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot F\right) \cdot \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot F\right), \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot A\right), \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 10.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \mathsf{*.f64}\left(C, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 10.7%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left(F \cdot A\right) \cdot \left(C \cdot C\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(F \cdot A\right) \cdot \left(C \cdot C\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(F \cdot A\right) \cdot C\right) \cdot C\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(F \cdot A\right) \cdot C\right) \cdot \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(F \cdot A\right) \cdot C\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(F \cdot \left(A \cdot C\right)\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A \cdot C\right)\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(C \cdot A\right)\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, A\right)\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-lowering-*.f64 14.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, A\right)\right), \mathsf{*.f64}\left(C, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 14.7%

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

   if 2.29999999999999991e-112 < B

   1. Initial program 20.4%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 38.3%

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

   5. Simplified 38.3%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 38.3%

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

   10. Simplified 38.3%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
   11. Step-by-step derivation

       1. unpow1/2 N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. sqrt-prod N/A

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

       5. unpow1/2 N/A

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

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

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

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

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

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

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

       9. unpow1/2 N/A

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

       10. sqrt-lowering-sqrt.f64 54.3%

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

   12. Applied egg-rr 54.3%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot -2} \cdot \sqrt{B}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 9.2e-108)
   (/ (sqrt (* (* F (* A C)) (* C -16.0))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.2e-108) {
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
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 <= 9.2d-108) then
        tmp = sqrt(((f * (a * c)) * (c * (-16.0d0)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = (((-2.0d0) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.2e-108) {
		tmp = Math.sqrt(((F * (A * C)) * (C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9.2e-108:
		tmp = math.sqrt(((F * (A * C)) * (C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 9.2e-108)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A * C)) * Float64(C * -16.0))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 9.2e-108)
		tmp = sqrt(((F * (A * C)) * (C * -16.0))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9.2e-108], N[(N[Sqrt[N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(C * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9.19999999999999983e-108

   1. Initial program 21.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 24.9%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(F \cdot A\right) \cdot \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot F\right) \cdot \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot F\right), \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot A\right), \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \left(C \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 10.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, A\right), \mathsf{*.f64}\left(C, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 10.7%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left(F \cdot A\right) \cdot \left(C \cdot C\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(F \cdot A\right) \cdot \left(C \cdot C\right)\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(F \cdot A\right) \cdot C\right) \cdot C\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(F \cdot A\right) \cdot C\right) \cdot \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(F \cdot A\right) \cdot C\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(F \cdot \left(A \cdot C\right)\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A \cdot C\right)\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(C \cdot A\right)\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, A\right)\right), \left(C \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-lowering-*.f64 14.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, A\right)\right), \mathsf{*.f64}\left(C, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 14.6%

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

   if 9.19999999999999983e-108 < B

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 38.7%

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

   5. Simplified 38.7%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 38.7%

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

   10. Simplified 38.7%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot -16\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.5e-108)
   (/ (sqrt (* -16.0 (* C (* F (* A C))))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.5e-108) {
		tmp = sqrt((-16.0 * (C * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
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 <= 4.5d-108) then
        tmp = sqrt(((-16.0d0) * (c * (f * (a * c))))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = (((-2.0d0) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.5e-108) {
		tmp = Math.sqrt((-16.0 * (C * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.5e-108:
		tmp = math.sqrt((-16.0 * (C * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.5e-108)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(C * Float64(F * Float64(A * C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.5e-108)
		tmp = sqrt((-16.0 * (C * (F * (A * C))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.5e-108], N[(N[Sqrt[N[(-16.0 * N[(C * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.4999999999999997e-108

   1. Initial program 21.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 24.9%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(\left(A \cdot C\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(F \cdot \left(A \cdot C\right)\right), C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(A \cdot C\right)\right), C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \left(C \cdot A\right)\right), C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 14.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, A\right)\right), C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 14.6%

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

   if 4.4999999999999997e-108 < B

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 38.7%

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

   5. Simplified 38.7%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 38.7%

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

   10. Simplified 38.7%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.3 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.25}{A \cdot C} \cdot \sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot C\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.3e-107)
   (* (/ 0.25 (* A C)) (sqrt (* -16.0 (* F (* C (* A C))))))
   (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.3e-107) {
		tmp = (0.25 / (A * C)) * sqrt((-16.0 * (F * (C * (A * C)))));
	} else {
		tmp = pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 2.3d-107) then
        tmp = (0.25d0 / (a * c)) * sqrt(((-16.0d0) * (f * (c * (a * c)))))
    else
        tmp = (((-2.0d0) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.3e-107) {
		tmp = (0.25 / (A * C)) * Math.sqrt((-16.0 * (F * (C * (A * C)))));
	} else {
		tmp = Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2.3e-107:
		tmp = (0.25 / (A * C)) * math.sqrt((-16.0 * (F * (C * (A * C)))))
	else:
		tmp = math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.3e-107)
		tmp = Float64(Float64(0.25 / Float64(A * C)) * sqrt(Float64(-16.0 * Float64(F * Float64(C * Float64(A * C))))));
	else
		tmp = Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.3e-107)
		tmp = (0.25 / (A * C)) * sqrt((-16.0 * (F * (C * (A * C)))));
	else
		tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.3e-107], N[(N[(0.25 / N[(A * C), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-16.0 * N[(F * N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.30000000000000003e-107

   1. Initial program 21.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 24.9%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-*r* N/A

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

      3. associate-/r/ N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right), \left(\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(4 \cdot \left(A \cdot C\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\left(4 \cdot A\right) \cdot C\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(C \cdot \left(4 \cdot A\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \left(4 \cdot A\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \left(A \cdot 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \color{blue}{F}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(F, \left(A \cdot \left(C \cdot C\right)\right)\right)\right)\right)\right) \]

      17. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(F, \left(\left(A \cdot C\right) \cdot C\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(F, \left(C \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

      20. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \left(C \cdot A\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 12.4%

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

   10. Taylor expanded in C around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1}{4}}{A \cdot C}\right)}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, \left(A \cdot C\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right)\right)\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, \left(C \cdot A\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \color{blue}{\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right)\right)}\right)\right)\right) \]

       3. *-lowering-*.f64 12.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(C, A\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \color{blue}{\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right)\right)}\right)\right)\right) \]

   12. Simplified 12.1%

       \[\leadsto \color{blue}{\frac{0.25}{C \cdot A}} \cdot \sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(C \cdot A\right)\right)\right)} \]

   if 2.30000000000000003e-107 < B

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 38.7%

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

   5. Simplified 38.7%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 38.9%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 38.7%

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

   10. Simplified 38.7%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{-107}:\\ \;\;\;\;\frac{0.25}{A \cdot C} \cdot \sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot C\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq 410:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{F \cdot \left(B\_m \cdot B\_m\right)}{0 - C}\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 410.0)
   (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m))
   (/ (pow (/ (* F (* B_m B_m)) (- 0.0 C)) 0.5) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 410.0) {
		tmp = pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = pow(((F * (B_m * B_m)) / (0.0 - C)), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 410.0d0) then
        tmp = (((-2.0d0) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
    else
        tmp = (((f * (b_m * b_m)) / (0.0d0 - c)) ** 0.5d0) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 410.0) {
		tmp = Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = Math.pow(((F * (B_m * B_m)) / (0.0 - C)), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= 410.0:
		tmp = math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)
	else:
		tmp = math.pow(((F * (B_m * B_m)) / (0.0 - C)), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 410.0)
		tmp = Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m));
	else
		tmp = Float64((Float64(Float64(F * Float64(B_m * B_m)) / Float64(0.0 - C)) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 410.0)
		tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
	else
		tmp = (((F * (B_m * B_m)) / (0.0 - C)) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 410.0], N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[(0.0 - C), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 410

   1. Initial program 24.6%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 15.3%

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

   5. Simplified 15.3%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 15.4%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

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

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

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

      2. *-lowering-*.f64 15.1%

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

   10. Simplified 15.1%

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

   if 410 < C

   1. Initial program 7.0%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

      12. hypot-lowering-hypot.f64 6.0%

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

   5. Simplified 6.0%

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

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

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

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

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

      4. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 6.0%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(0 - \frac{{B}^{2} \cdot F}{C}\right), \frac{1}{2}\right), B\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{{B}^{2} \cdot F}{C}\right)\right), \frac{1}{2}\right), B\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right), \frac{1}{2}\right), B\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), C\right)\right), \frac{1}{2}\right), B\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), C\right)\right), \frac{1}{2}\right), B\right)\right) \]

      7. *-lowering-*.f64 18.6%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), C\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. Simplified 18.6%

       \[\leadsto -\frac{{\color{blue}{\left(0 - \frac{\left(B \cdot B\right) \cdot F}{C}\right)}}^{0.5}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 410:\\ \;\;\;\;\frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{F \cdot \left(B \cdot B\right)}{0 - C}\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{{\left(-2 \cdot \left(B\_m \cdot F\right)\right)}^{0.5}}{0 - B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (/ (pow (* -2.0 (* B_m F)) 0.5) (- 0.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
}

Fortran

B_m = abs(b)
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) * (b_m * f)) ** 0.5d0) / (0.0d0 - b_m)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.pow((-2.0 * (B_m * F)), 0.5) / (0.0 - B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64((Float64(-2.0 * Float64(B_m * F)) ^ 0.5) / Float64(0.0 - B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = ((-2.0 * (B_m * F)) ^ 0.5) / (0.0 - B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Power[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.2%

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

3. Taylor expanded in A around 0

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. unpow2 N/A

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

   10. unpow2 N/A

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

   11. hypot-define N/A

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

   12. hypot-lowering-hypot.f64 13.5%

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

5. Simplified 13.5%

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

   1. associate-*l* N/A

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

   2. mul-1-neg N/A

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

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

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

   4. associate-*l/ N/A

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

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

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

7. Applied egg-rr 13.6%

   \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

8. Taylor expanded in C around 0

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

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

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

   2. *-lowering-*.f64 13.1%

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

10. Simplified 13.1%

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

11. Final simplification 13.1%

    \[\leadsto \frac{{\left(-2 \cdot \left(B \cdot F\right)\right)}^{0.5}}{0 - B} \]
12. Add Preprocessing

Alternative 13: 25.8% accurate, 5.8× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* B_m (* F -2.0))) (- 0.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((B_m * (F * -2.0))) / (0.0 - B_m);
}

Fortran

B_m = abs(b)
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((b_m * (f * (-2.0d0)))) / (0.0d0 - b_m)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((B_m * (F * -2.0))) / (0.0 - B_m);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((B_m * (F * -2.0))) / (0.0 - B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(B_m * Float64(F * -2.0))) / Float64(0.0 - B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((B_m * (F * -2.0))) / (0.0 - B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(B$95$m * N[(F * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.2%

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

3. Taylor expanded in A around 0

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. unpow2 N/A

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

   10. unpow2 N/A

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

   11. hypot-define N/A

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

   12. hypot-lowering-hypot.f64 13.5%

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

5. Simplified 13.5%

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

   1. associate-*l* N/A

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

   2. mul-1-neg N/A

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

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

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

   4. associate-*l/ N/A

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

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

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

7. Applied egg-rr 13.6%

   \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

8. Taylor expanded in C around 0

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

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

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

   2. *-lowering-*.f64 13.1%

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

10. Simplified 13.1%

    \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}}^{0.5}}{B} \]
11. Step-by-step derivation

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

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

    2. unpow1/2 N/A

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

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

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

    4. *-commutative N/A

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

    5. associate-*l* N/A

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

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

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

    7. *-lowering-*.f64 13.0%

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

12. Applied egg-rr 13.0%

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

13. Final simplification 13.0%

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

Reproduce

herbie shell --seed 2024138 
(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))))