ABCF->ab-angle a

Percentage Accurate: 19.4% → 42.9%

Time: 28.7s

Alternatives: 7

Speedup: 6.0×

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: 19.4% 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: 42.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in A around 0 5.8%

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

   1. mul-1-neg 5.8%

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

   2. *-commutative 5.8%

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

   3. distribute-rgt-neg-in 5.8%

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

   4. unpow2 5.8%

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

   5. unpow2 5.8%

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

   6. hypot-define 13.4%

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

5. Simplified 13.4%

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

   1. pow1/2 13.4%

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

   2. *-commutative 13.4%

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

   3. unpow-prod-down 19.0%

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

   4. pow1/2 19.0%

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

   5. pow1/2 19.0%

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

7. Applied egg-rr 19.0%

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

8. Final simplification 19.0%

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

Alternative 2: 35.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{1}{B\_m}} \cdot \sqrt{F \cdot 2} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (- (* (sqrt (/ 1.0 B_m)) (sqrt (* F 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -(sqrt((1.0 / B_m)) * sqrt((F * 2.0)));
}

Fortran

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

Java

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

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -(math.sqrt((1.0 / B_m)) * math.sqrt((F * 2.0)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-Float64(sqrt(Float64(1.0 / B_m)) * sqrt(Float64(F * 2.0))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -(sqrt((1.0 / B_m)) * sqrt((F * 2.0)));
end

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in B around inf 12.8%

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

   1. mul-1-neg 12.8%

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

   2. *-commutative 12.8%

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

   3. distribute-rgt-neg-in 12.8%

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

5. Simplified 12.8%

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

   1. pow1 12.8%

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

   2. distribute-rgt-neg-out 12.8%

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

   3. pow1/2 12.8%

      \[\leadsto {\left(-\color{blue}{{2}^{0.5}} \cdot \sqrt{\frac{F}{B}}\right)}^{1} \]

   4. pow1/2 13.1%

      \[\leadsto {\left(-{2}^{0.5} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{1} \]

   5. pow-prod-down 13.1%

      \[\leadsto {\left(-\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{0.5}}\right)}^{1} \]

7. Applied egg-rr 13.1%

   \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 13.1%

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

   2. unpow1/2 12.9%

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

   3. associate-*r/ 12.9%

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

9. Simplified 12.9%

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

    1. pow1/2 13.1%

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

    2. div-inv 13.1%

       \[\leadsto -{\color{blue}{\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}}^{0.5} \]

    3. unpow-prod-down 16.3%

       \[\leadsto -\color{blue}{{\left(2 \cdot F\right)}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}} \]

    4. pow1/2 16.3%

       \[\leadsto -\color{blue}{\sqrt{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{0.5} \]

    5. metadata-eval 16.3%

       \[\leadsto -\sqrt{\color{blue}{{2}^{1}} \cdot F} \cdot {\left(\frac{1}{B}\right)}^{0.5} \]

    6. metadata-eval 16.3%

       \[\leadsto -\sqrt{{2}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot F} \cdot {\left(\frac{1}{B}\right)}^{0.5} \]

    7. sqrt-pow2 16.2%

       \[\leadsto -\sqrt{\color{blue}{{\left(\sqrt{2}\right)}^{2}} \cdot F} \cdot {\left(\frac{1}{B}\right)}^{0.5} \]

    8. *-commutative 16.2%

       \[\leadsto -\sqrt{\color{blue}{F \cdot {\left(\sqrt{2}\right)}^{2}}} \cdot {\left(\frac{1}{B}\right)}^{0.5} \]

    9. sqrt-pow2 16.3%

       \[\leadsto -\sqrt{F \cdot \color{blue}{{2}^{\left(\frac{2}{2}\right)}}} \cdot {\left(\frac{1}{B}\right)}^{0.5} \]

    10. metadata-eval 16.3%

        \[\leadsto -\sqrt{F \cdot {2}^{\color{blue}{1}}} \cdot {\left(\frac{1}{B}\right)}^{0.5} \]

    11. metadata-eval 16.3%

        \[\leadsto -\sqrt{F \cdot \color{blue}{2}} \cdot {\left(\frac{1}{B}\right)}^{0.5} \]

11. Applied egg-rr 16.3%

    \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {\left(\frac{1}{B}\right)}^{0.5}} \]
12. Step-by-step derivation

    1. unpow1/2 16.3%

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

    2. *-commutative 16.3%

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

13. Simplified 16.3%

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

14. Final simplification 16.3%

    \[\leadsto -\sqrt{\frac{1}{B}} \cdot \sqrt{F \cdot 2} \]
15. Add Preprocessing

Alternative 3: 35.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in B around inf 12.8%

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

   1. mul-1-neg 12.8%

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

   2. *-commutative 12.8%

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

   3. distribute-rgt-neg-in 12.8%

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

5. Simplified 12.8%

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

   1. pow1 12.8%

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

   2. distribute-rgt-neg-out 12.8%

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

   3. pow1/2 12.8%

      \[\leadsto {\left(-\color{blue}{{2}^{0.5}} \cdot \sqrt{\frac{F}{B}}\right)}^{1} \]

   4. pow1/2 13.1%

      \[\leadsto {\left(-{2}^{0.5} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{1} \]

   5. pow-prod-down 13.1%

      \[\leadsto {\left(-\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{0.5}}\right)}^{1} \]

7. Applied egg-rr 13.1%

   \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 13.1%

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

   2. unpow1/2 12.9%

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

   3. associate-*r/ 12.9%

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

9. Simplified 12.9%

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

    1. sqrt-div 16.3%

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

    2. metadata-eval 16.3%

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

    3. metadata-eval 16.3%

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

    4. sqrt-pow2 16.2%

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

    5. *-commutative 16.2%

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

    6. sqrt-pow2 16.3%

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

    7. metadata-eval 16.3%

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

    8. metadata-eval 16.3%

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

11. Applied egg-rr 16.3%

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

12. Final simplification 16.3%

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

Alternative 4: 27.1% accurate, 5.9× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (pow (* 2.0 (/ F B_m)) 0.5)))

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -((2.0d0 * (f / b_m)) ** 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in B around inf 12.8%

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

   1. mul-1-neg 12.8%

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

   2. *-commutative 12.8%

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

   3. distribute-rgt-neg-in 12.8%

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

5. Simplified 12.8%

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

   1. distribute-rgt-neg-out 12.8%

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

   2. pow1/2 12.8%

      \[\leadsto -\color{blue}{{2}^{0.5}} \cdot \sqrt{\frac{F}{B}} \]

   3. pow1/2 13.1%

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

   4. pow-prod-down 13.1%

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

7. Applied egg-rr 13.1%

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

8. Final simplification 13.1%

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

Alternative 5: 27.1% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in B around inf 12.8%

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

   1. mul-1-neg 12.8%

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

   2. *-commutative 12.8%

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

   3. distribute-rgt-neg-in 12.8%

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

5. Simplified 12.8%

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

   1. pow1 12.8%

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

   2. distribute-rgt-neg-out 12.8%

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

   3. pow1/2 12.8%

      \[\leadsto {\left(-\color{blue}{{2}^{0.5}} \cdot \sqrt{\frac{F}{B}}\right)}^{1} \]

   4. pow1/2 13.1%

      \[\leadsto {\left(-{2}^{0.5} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{1} \]

   5. pow-prod-down 13.1%

      \[\leadsto {\left(-\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{0.5}}\right)}^{1} \]

7. Applied egg-rr 13.1%

   \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 13.1%

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

   2. unpow1/2 12.9%

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

   3. associate-*r/ 12.9%

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

9. Simplified 12.9%

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

10. Final simplification 12.9%

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

Alternative 6: 2.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in B around inf 12.8%

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

   1. mul-1-neg 12.8%

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

   2. *-commutative 12.8%

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

   3. distribute-rgt-neg-in 12.8%

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

5. Simplified 12.8%

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

   1. add-log-exp 3.4%

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

7. Applied egg-rr 3.4%

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

   1. metadata-eval 3.4%

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

   2. metadata-eval 3.4%

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

   3. sqrt-pow2 3.4%

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

   4. add-sqr-sqrt 1.4%

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

   5. sqrt-unprod 1.8%

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

   6. sqr-neg 1.8%

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

   7. rem-log-exp 1.8%

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

   8. rem-log-exp 2.0%

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

   9. add-sqr-sqrt 2.0%

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

   10. sqrt-prod 2.0%

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

   11. sqrt-pow2 2.0%

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

   12. metadata-eval 2.0%

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

   13. metadata-eval 2.0%

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

9. Applied egg-rr 2.0%

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

10. Final simplification 2.0%

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

Alternative 7: 2.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in B around inf 12.8%

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

   1. mul-1-neg 12.8%

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

   2. *-commutative 12.8%

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

   3. distribute-rgt-neg-in 12.8%

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

5. Simplified 12.8%

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

   1. add-log-exp 3.4%

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

7. Applied egg-rr 3.4%

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

   1. metadata-eval 3.4%

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

   2. metadata-eval 3.4%

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

   3. sqrt-pow2 3.4%

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

   4. add-sqr-sqrt 1.4%

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

   5. sqrt-unprod 1.8%

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

   6. sqr-neg 1.8%

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

   7. rem-log-exp 1.8%

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

   8. rem-log-exp 2.0%

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

   9. add-sqr-sqrt 2.0%

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

   10. sqrt-prod 2.0%

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

   11. sqrt-pow2 2.0%

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

   12. metadata-eval 2.0%

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

   13. metadata-eval 2.0%

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

9. Applied egg-rr 2.0%

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

    1. *-un-lft-identity 2.0%

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

11. Applied egg-rr 2.0%

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

    1. *-lft-identity 2.0%

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

    2. associate-*r/ 2.0%

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

    3. *-commutative 2.0%

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

13. Simplified 2.0%

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

14. Final simplification 2.0%

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

Reproduce

herbie shell --seed 2024066 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))