ABCF->ab-angle b

Percentage Accurate: 19.1% → 45.1%

Time: 22.8s

Alternatives: 10

Speedup: 5.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.1% 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: 45.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 5.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)}{B}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+260}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B} \cdot \frac{C - \mathsf{hypot}\left(B, C\right)}{B}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 5.5e-61)
     (/ (- (sqrt (* 2.0 (* (* t_0 F) (* 2.0 A))))) t_0)
     (if (<= B 5.5e+39)
       (/ (* (sqrt 2.0) (- (sqrt (* F (* -0.5 (/ (* B B) C)))))) B)
       (if (<= B 9.5e+260)
         (/ (- (sqrt (* 2.0 (* F (- A (hypot A B)))))) B)
         (- (sqrt (* (/ (* 2.0 F) B) (/ (- C (hypot B C)) B)))))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 5.5e-61) {
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else if (B <= 5.5e+39) {
		tmp = (sqrt(2.0) * -sqrt((F * (-0.5 * ((B * B) / C))))) / B;
	} else if (B <= 9.5e+260) {
		tmp = -sqrt((2.0 * (F * (A - hypot(A, B))))) / B;
	} else {
		tmp = -sqrt((((2.0 * F) / B) * ((C - hypot(B, C)) / B)));
	}
	return tmp;
}

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 5.5e-61) {
		tmp = -Math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	} else if (B <= 5.5e+39) {
		tmp = (Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 * ((B * B) / C))))) / B;
	} else if (B <= 9.5e+260) {
		tmp = -Math.sqrt((2.0 * (F * (A - Math.hypot(A, B))))) / B;
	} else {
		tmp = -Math.sqrt((((2.0 * F) / B) * ((C - Math.hypot(B, C)) / B)));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 5.5e-61:
		tmp = -math.sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0
	elif B <= 5.5e+39:
		tmp = (math.sqrt(2.0) * -math.sqrt((F * (-0.5 * ((B * B) / C))))) / B
	elif B <= 9.5e+260:
		tmp = -math.sqrt((2.0 * (F * (A - math.hypot(A, B))))) / B
	else:
		tmp = -math.sqrt((((2.0 * F) / B) * ((C - math.hypot(B, C)) / B)))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 5.5e-61)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(2.0 * A))))) / t_0);
	elseif (B <= 5.5e+39)
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B * B) / C)))))) / B);
	elseif (B <= 9.5e+260)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(A, B)))))) / B);
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(2.0 * F) / B) * Float64(Float64(C - hypot(B, C)) / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 5.5e-61)
		tmp = -sqrt((2.0 * ((t_0 * F) * (2.0 * A)))) / t_0;
	elseif (B <= 5.5e+39)
		tmp = (sqrt(2.0) * -sqrt((F * (-0.5 * ((B * B) / C))))) / B;
	elseif (B <= 9.5e+260)
		tmp = -sqrt((2.0 * (F * (A - hypot(A, B))))) / B;
	else
		tmp = -sqrt((((2.0 * F) / B) * ((C - hypot(B, C)) / B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 5.5e-61], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 5.5e+39], N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 9.5e+260], N[((-N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision], (-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] / B), $MachinePrecision] * N[(N[(C - N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]]

Derivation

1. Split input into 4 regimes

2. if B < 5.4999999999999997e-61

   1. Initial program 19.6%

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

   3. Simplified 19.5%

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

   4. Taylor expanded in A around -inf 19.6%

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

   if 5.4999999999999997e-61 < B < 5.4999999999999997e39

   1. Initial program 21.0%

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

   3. Simplified 21.0%

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

      1. unpow2 21.0%

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

      2. hypot-udef 21.0%

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

      3. add-exp-log 20.4%

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

   5. Applied egg-rr 20.4%

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

   6. Taylor expanded in A around 0 16.1%

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

      1. mul-1-neg 16.1%

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

      2. associate-*l/ 16.1%

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

      3. *-commutative 16.1%

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

      4. unpow2 16.1%

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

      5. unpow2 16.1%

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

      6. hypot-def 16.5%

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

   8. Simplified 16.5%

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

   9. Taylor expanded in C around inf 13.7%

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

       1. unpow2 13.7%

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

   11. Simplified 13.7%

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

   if 5.4999999999999997e39 < B < 9.5000000000000004e260

   1. Initial program 30.1%

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

   3. Simplified 30.1%

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

   4. Taylor expanded in C around 0 29.3%

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

      1. mul-1-neg 29.3%

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

      2. +-commutative 29.3%

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

      3. unpow2 29.3%

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

      4. unpow2 29.3%

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

      5. hypot-def 73.4%

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

   6. Simplified 73.4%

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

      1. associate-*l/ 73.4%

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

   8. Applied egg-rr 73.4%

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

      1. sqrt-unprod 73.5%

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

   10. Applied egg-rr 73.5%

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

   if 9.5000000000000004e260 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. unpow2 0.0%

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

      2. hypot-udef 0.0%

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

      3. add-exp-log 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in A around 0 2.4%

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

      1. mul-1-neg 2.4%

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

      2. associate-*l/ 2.4%

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

      3. *-commutative 2.4%

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

      4. unpow2 2.4%

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

      5. unpow2 2.4%

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

      6. hypot-def 39.6%

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

   8. Simplified 39.6%

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

      1. add-sqr-sqrt 39.5%

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

      2. sqrt-unprod 15.7%

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

      3. frac-times 2.5%

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

      4. sqrt-unprod 2.5%

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

      5. sqrt-unprod 2.5%

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

      6. add-sqr-sqrt 2.5%

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

   10. Applied egg-rr 2.5%

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

       1. associate-*r* 2.5%

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

       2. times-frac 75.7%

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

   12. Simplified 75.7%

       \[\leadsto -\color{blue}{\sqrt{\frac{2 \cdot F}{B} \cdot \frac{C - \mathsf{hypot}\left(B, C\right)}{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{C}\right)}\right)}{B}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+260}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B} \cdot \frac{C - \mathsf{hypot}\left(B, C\right)}{B}}\\ \end{array} \]

Alternative 2: 47.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ \mathbf{if}\;B \leq 1.45 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+260}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B} \cdot \frac{C - \mathsf{hypot}\left(B, C\right)}{B}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* A C) -4.0))))
   (if (<= B 1.45e+25)
     (/ (- (sqrt (* 2.0 (* t_0 (* F (* 2.0 A)))))) t_0)
     (if (<= B 7.6e+260)
       (/ (- (sqrt (* 2.0 (* F (- A (hypot A B)))))) B)
       (- (sqrt (* (/ (* 2.0 F) B) (/ (- C (hypot B C)) B))))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 1.45e+25) {
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else if (B <= 7.6e+260) {
		tmp = -sqrt((2.0 * (F * (A - hypot(A, B))))) / B;
	} else {
		tmp = -sqrt((((2.0 * F) / B) * ((C - hypot(B, C)) / B)));
	}
	return tmp;
}

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 1.45e+25) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else if (B <= 7.6e+260) {
		tmp = -Math.sqrt((2.0 * (F * (A - Math.hypot(A, B))))) / B;
	} else {
		tmp = -Math.sqrt((((2.0 * F) / B) * ((C - Math.hypot(B, C)) / B)));
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + ((A * C) * -4.0)
	tmp = 0
	if B <= 1.45e+25:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0
	elif B <= 7.6e+260:
		tmp = -math.sqrt((2.0 * (F * (A - math.hypot(A, B))))) / B
	else:
		tmp = -math.sqrt((((2.0 * F) / B) * ((C - math.hypot(B, C)) / B)))
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B <= 1.45e+25)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(2.0 * A)))))) / t_0);
	elseif (B <= 7.6e+260)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(A, B)))))) / B);
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(2.0 * F) / B) * Float64(Float64(C - hypot(B, C)) / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((A * C) * -4.0);
	tmp = 0.0;
	if (B <= 1.45e+25)
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	elseif (B <= 7.6e+260)
		tmp = -sqrt((2.0 * (F * (A - hypot(A, B))))) / B;
	else
		tmp = -sqrt((((2.0 * F) / B) * ((C - hypot(B, C)) / B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.45e+25], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 7.6e+260], N[((-N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision], (-N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] / B), $MachinePrecision] * N[(N[(C - N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.44999999999999995e25

   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. Step-by-step derivation

   3. Simplified 19.5%

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

   4. Taylor expanded in A around -inf 19.0%

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

      1. distribute-frac-neg 19.0%

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

      2. associate-*l* 18.6%

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

      3. cancel-sign-sub-inv 18.6%

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

      4. metadata-eval 18.6%

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

      5. *-commutative 18.6%

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

      6. cancel-sign-sub-inv 18.6%

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

      7. metadata-eval 18.6%

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

   6. Applied egg-rr 18.6%

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

   if 1.44999999999999995e25 < B < 7.5999999999999995e260

   1. Initial program 30.6%

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

   3. Simplified 30.6%

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

   4. Taylor expanded in C around 0 28.1%

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

      1. mul-1-neg 28.1%

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

      2. +-commutative 28.1%

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

      3. unpow2 28.1%

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

      4. unpow2 28.1%

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

      5. hypot-def 68.2%

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

   6. Simplified 68.2%

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

      1. associate-*l/ 68.2%

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

   8. Applied egg-rr 68.2%

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

      1. sqrt-unprod 68.3%

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

   10. Applied egg-rr 68.3%

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

   if 7.5999999999999995e260 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

      1. unpow2 0.0%

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

      2. hypot-udef 0.0%

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

      3. add-exp-log 0.0%

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

   5. Applied egg-rr 0.0%

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

   6. Taylor expanded in A around 0 2.4%

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

      1. mul-1-neg 2.4%

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

      2. associate-*l/ 2.4%

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

      3. *-commutative 2.4%

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

      4. unpow2 2.4%

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

      5. unpow2 2.4%

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

      6. hypot-def 39.6%

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

   8. Simplified 39.6%

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

      1. add-sqr-sqrt 39.5%

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

      2. sqrt-unprod 15.7%

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

      3. frac-times 2.5%

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

      4. sqrt-unprod 2.5%

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

      5. sqrt-unprod 2.5%

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

      6. add-sqr-sqrt 2.5%

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

   10. Applied egg-rr 2.5%

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

       1. associate-*r* 2.5%

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

       2. times-frac 75.7%

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

   12. Simplified 75.7%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.45 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B + \left(A \cdot C\right) \cdot -4}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+260}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B} \cdot \frac{C - \mathsf{hypot}\left(B, C\right)}{B}}\\ \end{array} \]

Alternative 3: 47.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ \mathbf{if}\;B \leq 3.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* A C) -4.0))))
   (if (<= B 3.6e+23)
     (/ (- (sqrt (* 2.0 (* t_0 (* F (* 2.0 A)))))) t_0)
     (/ (- (sqrt (* 2.0 (* F (- A (hypot A B)))))) B))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 3.6e+23) {
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else {
		tmp = -sqrt((2.0 * (F * (A - hypot(A, B))))) / B;
	}
	return tmp;
}

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 3.6e+23) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else {
		tmp = -Math.sqrt((2.0 * (F * (A - Math.hypot(A, B))))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + ((A * C) * -4.0)
	tmp = 0
	if B <= 3.6e+23:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0
	else:
		tmp = -math.sqrt((2.0 * (F * (A - math.hypot(A, B))))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B <= 3.6e+23)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(2.0 * A)))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(A, B)))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((A * C) * -4.0);
	tmp = 0.0;
	if (B <= 3.6e+23)
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	else
		tmp = -sqrt((2.0 * (F * (A - hypot(A, B))))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 3.6e+23], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.5999999999999998e23

   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. Step-by-step derivation

   3. Simplified 19.5%

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

   4. Taylor expanded in A around -inf 19.0%

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

      1. distribute-frac-neg 19.0%

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

      2. associate-*l* 18.6%

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

      3. cancel-sign-sub-inv 18.6%

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

      4. metadata-eval 18.6%

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

      5. *-commutative 18.6%

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

      6. cancel-sign-sub-inv 18.6%

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

      7. metadata-eval 18.6%

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

   6. Applied egg-rr 18.6%

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

   if 3.5999999999999998e23 < B

   1. Initial program 21.9%

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

   3. Simplified 21.9%

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

   4. Taylor expanded in C around 0 20.9%

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

      1. mul-1-neg 20.9%

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

      2. +-commutative 20.9%

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

      3. unpow2 20.9%

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

      4. unpow2 20.9%

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

      5. hypot-def 60.1%

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

   6. Simplified 60.1%

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

      1. associate-*l/ 60.1%

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

   8. Applied egg-rr 60.1%

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

      1. sqrt-unprod 60.3%

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

   10. Applied egg-rr 60.3%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B + \left(A \cdot C\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 4: 44.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ \mathbf{if}\;B \leq 8.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - B\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* A C) -4.0))))
   (if (<= B 8.2e+23)
     (/ (- (sqrt (* 2.0 (* t_0 (* F (* 2.0 A)))))) t_0)
     (/ (- (sqrt (* 2.0 (* F (- A B))))) B))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 8.2e+23) {
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else {
		tmp = -sqrt((2.0 * (F * (A - B)))) / B;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    real(8) :: tmp
    t_0 = (b * b) + ((a * c) * (-4.0d0))
    if (b <= 8.2d+23) then
        tmp = -sqrt((2.0d0 * (t_0 * (f * (2.0d0 * a))))) / t_0
    else
        tmp = -sqrt((2.0d0 * (f * (a - b)))) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((A * C) * -4.0);
	double tmp;
	if (B <= 8.2e+23) {
		tmp = -Math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	} else {
		tmp = -Math.sqrt((2.0 * (F * (A - B)))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + ((A * C) * -4.0)
	tmp = 0
	if B <= 8.2e+23:
		tmp = -math.sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0
	else:
		tmp = -math.sqrt((2.0 * (F * (A - B)))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B <= 8.2e+23)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(2.0 * A)))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(A - B))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((A * C) * -4.0);
	tmp = 0.0;
	if (B <= 8.2e+23)
		tmp = -sqrt((2.0 * (t_0 * (F * (2.0 * A))))) / t_0;
	else
		tmp = -sqrt((2.0 * (F * (A - B)))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 8.2e+23], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 8.19999999999999992e23

   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. Step-by-step derivation

   3. Simplified 19.5%

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

   4. Taylor expanded in A around -inf 19.0%

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

      1. distribute-frac-neg 19.0%

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

      2. associate-*l* 18.6%

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

      3. cancel-sign-sub-inv 18.6%

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

      4. metadata-eval 18.6%

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

      5. *-commutative 18.6%

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

      6. cancel-sign-sub-inv 18.6%

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

      7. metadata-eval 18.6%

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

   6. Applied egg-rr 18.6%

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

   if 8.19999999999999992e23 < B

   1. Initial program 21.9%

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

   3. Simplified 21.9%

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

   4. Taylor expanded in C around 0 20.9%

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

      1. mul-1-neg 20.9%

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

      2. +-commutative 20.9%

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

      3. unpow2 20.9%

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

      4. unpow2 20.9%

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

      5. hypot-def 60.1%

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

   6. Simplified 60.1%

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

      1. associate-*l/ 60.1%

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

   8. Applied egg-rr 60.1%

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

      1. sqrt-unprod 60.3%

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

   10. Applied egg-rr 60.3%

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

   11. Taylor expanded in A around 0 59.1%

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

       1. mul-1-neg 59.1%

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

       2. unsub-neg 59.1%

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

   13. Simplified 59.1%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{B \cdot B + \left(A \cdot C\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - B\right)\right)}}{B}\\ \end{array} \]

Alternative 5: 42.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 1.38 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(A \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - B\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 1.38e-5)
   (/
    (- (sqrt (* 2.0 (* -8.0 (* A (* (* A C) F))))))
    (- (* B B) (* 4.0 (* A C))))
   (/ (- (sqrt (* 2.0 (* F (- A B))))) B)))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 1.38e-5) {
		tmp = -sqrt((2.0 * (-8.0 * (A * ((A * C) * F))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -sqrt((2.0 * (F * (A - B)))) / B;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (b <= 1.38d-5) then
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (a * ((a * c) * f))))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = -sqrt((2.0d0 * (f * (a - b)))) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 1.38e-5) {
		tmp = -Math.sqrt((2.0 * (-8.0 * (A * ((A * C) * F))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -Math.sqrt((2.0 * (F * (A - B)))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 1.38e-5:
		tmp = -math.sqrt((2.0 * (-8.0 * (A * ((A * C) * F))))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -math.sqrt((2.0 * (F * (A - B)))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 1.38e-5)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(A * Float64(Float64(A * C) * F)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(A - B))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 1.38e-5)
		tmp = -sqrt((2.0 * (-8.0 * (A * ((A * C) * F))))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -sqrt((2.0 * (F * (A - B)))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 1.38e-5], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(A * N[(N[(A * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.38e-5

   1. Initial program 19.9%

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

   3. Simplified 19.9%

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

   4. Taylor expanded in A around -inf 19.4%

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

   5. Taylor expanded in B around 0 11.5%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-commutative 11.5%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left({A}^{2} \cdot \color{blue}{\left(F \cdot C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 11.5%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\color{blue}{\left(A \cdot A\right)} \cdot \left(F \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. associate-*l* 15.2%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(A \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 15.2%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(A \cdot \left(A \cdot \color{blue}{\left(C \cdot F\right)}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. associate-*r* 18.9%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(A \cdot \color{blue}{\left(\left(A \cdot C\right) \cdot F\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 18.9%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(A \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.38e-5 < B

   1. Initial program 20.0%

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

   3. Simplified 20.0%

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

   4. Taylor expanded in C around 0 19.2%

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

      1. mul-1-neg 19.2%

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

      2. +-commutative 19.2%

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

      3. unpow2 19.2%

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

      4. unpow2 19.2%

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

      5. hypot-def 51.6%

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

   6. Simplified 51.6%

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

      1. associate-*l/ 51.6%

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

   8. Applied egg-rr 51.6%

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

      1. sqrt-unprod 51.8%

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

   10. Applied egg-rr 51.8%

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

   11. Taylor expanded in A around 0 50.3%

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

       1. mul-1-neg 50.3%

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

       2. unsub-neg 50.3%

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

   13. Simplified 50.3%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.38 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(A \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - B\right)\right)}}{B}\\ \end{array} \]

Alternative 6: 28.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.3 \cdot 10^{+185}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B} \cdot \frac{A}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - B\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= A -1.3e+185)
   (* -2.0 (sqrt (* (/ F B) (/ A B))))
   (/ (- (sqrt (* 2.0 (* F (- A B))))) B)))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -1.3e+185) {
		tmp = -2.0 * sqrt(((F / B) * (A / B)));
	} else {
		tmp = -sqrt((2.0 * (F * (A - B)))) / B;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (a <= (-1.3d+185)) then
        tmp = (-2.0d0) * sqrt(((f / b) * (a / b)))
    else
        tmp = -sqrt((2.0d0 * (f * (a - b)))) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -1.3e+185) {
		tmp = -2.0 * Math.sqrt(((F / B) * (A / B)));
	} else {
		tmp = -Math.sqrt((2.0 * (F * (A - B)))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if A <= -1.3e+185:
		tmp = -2.0 * math.sqrt(((F / B) * (A / B)))
	else:
		tmp = -math.sqrt((2.0 * (F * (A - B)))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (A <= -1.3e+185)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(F / B) * Float64(A / B))));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(A - B))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= -1.3e+185)
		tmp = -2.0 * sqrt(((F / B) * (A / B)));
	else
		tmp = -sqrt((2.0 * (F * (A - B)))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[A, -1.3e+185], N[(-2.0 * N[Sqrt[N[(N[(F / B), $MachinePrecision] * N[(A / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * N[(A - B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.3e185

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   4. Taylor expanded in A around -inf 20.4%

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

   5. Taylor expanded in B around inf 5.0%

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

      1. associate-*r/ 5.0%

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

      2. *-rgt-identity 5.0%

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

      3. *-commutative 5.0%

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

   7. Simplified 5.0%

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

      1. add-sqr-sqrt 4.5%

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

      2. sqrt-unprod 6.6%

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

      3. frac-times 2.9%

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

      4. add-sqr-sqrt 2.9%

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

      5. *-commutative 2.9%

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

   9. Applied egg-rr 2.9%

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

       1. *-commutative 2.9%

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

       2. times-frac 29.4%

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

   11. Simplified 29.4%

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

   if -1.3e185 < A

   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

   3. Simplified 21.5%

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

   4. Taylor expanded in C around 0 8.4%

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

      1. mul-1-neg 8.4%

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

      2. +-commutative 8.4%

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

      3. unpow2 8.4%

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

      4. unpow2 8.4%

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

      5. hypot-def 16.4%

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

   6. Simplified 16.4%

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

      1. associate-*l/ 16.3%

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

   8. Applied egg-rr 16.3%

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

      1. sqrt-unprod 16.4%

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

   10. Applied egg-rr 16.4%

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

   11. Taylor expanded in A around 0 14.0%

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

       1. mul-1-neg 14.0%

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

       2. unsub-neg 14.0%

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

   13. Simplified 14.0%

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

4. Final simplification 15.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.3 \cdot 10^{+185}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B} \cdot \frac{A}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - B\right)\right)}}{B}\\ \end{array} \]

Alternative 7: 28.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{+199}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B} \cdot \frac{A}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(-B\right)\right)}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= A -3.2e+199)
   (* -2.0 (sqrt (* (/ F B) (/ A B))))
   (/ (- (sqrt (* 2.0 (* F (- B))))) B)))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -3.2e+199) {
		tmp = -2.0 * sqrt(((F / B) * (A / B)));
	} else {
		tmp = -sqrt((2.0 * (F * -B))) / B;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (a <= (-3.2d+199)) then
        tmp = (-2.0d0) * sqrt(((f / b) * (a / b)))
    else
        tmp = -sqrt((2.0d0 * (f * -b))) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -3.2e+199) {
		tmp = -2.0 * Math.sqrt(((F / B) * (A / B)));
	} else {
		tmp = -Math.sqrt((2.0 * (F * -B))) / B;
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if A <= -3.2e+199:
		tmp = -2.0 * math.sqrt(((F / B) * (A / B)))
	else:
		tmp = -math.sqrt((2.0 * (F * -B))) / B
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (A <= -3.2e+199)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(F / B) * Float64(A / B))));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(-B))))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= -3.2e+199)
		tmp = -2.0 * sqrt(((F / B) * (A / B)));
	else
		tmp = -sqrt((2.0 * (F * -B))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[A, -3.2e+199], N[(-2.0 * N[Sqrt[N[(N[(F / B), $MachinePrecision] * N[(A / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * (-B)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -3.20000000000000006e199

   1. Initial program 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in A around -inf 17.3%

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

   5. Taylor expanded in B around inf 5.0%

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

      1. associate-*r/ 5.0%

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

      2. *-rgt-identity 5.0%

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

      3. *-commutative 5.0%

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

   7. Simplified 5.0%

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

      1. add-sqr-sqrt 4.4%

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

      2. sqrt-unprod 6.8%

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

      3. frac-times 2.7%

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

      4. add-sqr-sqrt 2.7%

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

      5. *-commutative 2.7%

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

   9. Applied egg-rr 2.7%

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

       1. *-commutative 2.7%

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

       2. times-frac 32.1%

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

   11. Simplified 32.1%

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

   if -3.20000000000000006e199 < A

   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

   3. Simplified 21.4%

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

   4. Taylor expanded in C around 0 8.3%

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

      1. mul-1-neg 8.3%

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

      2. +-commutative 8.3%

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

      3. unpow2 8.3%

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

      4. unpow2 8.3%

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

      5. hypot-def 16.3%

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

   6. Simplified 16.3%

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

      1. associate-*l/ 16.2%

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

   8. Applied egg-rr 16.2%

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

      1. sqrt-unprod 16.3%

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

   10. Applied egg-rr 16.3%

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

   11. Taylor expanded in A around 0 14.6%

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

       1. associate-*r* 14.6%

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

       2. mul-1-neg 14.6%

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

   13. Simplified 14.6%

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

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{+199}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B} \cdot \frac{A}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(-B\right)\right)}}{B}\\ \end{array} \]

Alternative 8: 26.7% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    code = -sqrt((2.0d0 * (f * -b))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.9%

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

3. Simplified 19.9%

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

4. Taylor expanded in C around 0 7.8%

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

   1. mul-1-neg 7.8%

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

   2. +-commutative 7.8%

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

   3. unpow2 7.8%

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

   4. unpow2 7.8%

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

   5. hypot-def 15.4%

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

6. Simplified 15.4%

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

   1. associate-*l/ 15.4%

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

8. Applied egg-rr 15.4%

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

   1. sqrt-unprod 15.5%

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

10. Applied egg-rr 15.5%

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

11. Taylor expanded in A around 0 13.7%

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

    1. associate-*r* 13.7%

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

    2. mul-1-neg 13.7%

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

13. Simplified 13.7%

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

14. Final simplification 13.7%

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

Alternative 9: 8.8% accurate, 5.9× speedup

Math

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

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (pow (* A F) 0.5) B)))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	return -2.0 * (pow((A * F), 0.5) / B);
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    code = (-2.0d0) * (((a * f) ** 0.5d0) / b)
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.pow((A * F), 0.5) / B);
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	return -2.0 * (math.pow((A * F), 0.5) / B)

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	return Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B))
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (((A * F) ^ 0.5) / B);
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.9%

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

3. Simplified 19.9%

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

4. Taylor expanded in A around -inf 15.7%

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

5. Taylor expanded in B around inf 2.9%

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

   1. associate-*r/ 2.9%

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

   2. *-rgt-identity 2.9%

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

   3. *-commutative 2.9%

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

7. Simplified 2.9%

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

   1. pow1/2 3.0%

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

   2. *-commutative 3.0%

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

9. Applied egg-rr 3.0%

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

10. Final simplification 3.0%

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

Alternative 10: 8.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    code = (-2.0d0) * (sqrt((a * f)) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.9%

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

3. Simplified 19.9%

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

4. Taylor expanded in A around -inf 15.7%

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

5. Taylor expanded in B around inf 2.9%

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

   1. associate-*r/ 2.9%

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

   2. *-rgt-identity 2.9%

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

   3. *-commutative 2.9%

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

7. Simplified 2.9%

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

8. Final simplification 2.9%

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

Reproduce

herbie shell --seed 2023214 
(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))))