ABCF->ab-angle a

Percentage Accurate: 18.8% → 59.0%

Time: 24.0s

Alternatives: 11

Speedup: 6.0×

• Could not converge on a ground truth

  1. reg0 = (bf "-1.253602212582965399895514121525053747694e176")
  2. reg1 = (bf "5.345666161401996185557337466919222234376e-167")
  3. reg2 = (bf "2.430212734071377974792088522991892776987e272")
  4. reg3 = (bf "7.0341362358626362779766363299385703039e-241")

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.8% 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: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := t\_0 \cdot F\\ t_2 := -t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot 4\right) \cdot t\_1}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+113}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_1} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (* t_0 F)) (t_2 (- t_0)))
   (if (<= (pow B_m 2.0) 2e-40)
     (/ (sqrt (* (* C 4.0) t_1)) t_2)
     (if (<= (pow B_m 2.0) 1e+113)
       (/ (* (sqrt (* 2.0 t_1)) (sqrt (+ A (+ C (hypot (- A C) B_m))))) t_2)
       (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = t_0 * F;
	double t_2 = -t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-40) {
		tmp = sqrt(((C * 4.0) * t_1)) / t_2;
	} else if (pow(B_m, 2.0) <= 1e+113) {
		tmp = (sqrt((2.0 * t_1)) * sqrt((A + (C + hypot((A - C), B_m))))) / t_2;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(t_0 * F)
	t_2 = Float64(-t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-40)
		tmp = Float64(sqrt(Float64(Float64(C * 4.0) * t_1)) / t_2);
	elseif ((B_m ^ 2.0) <= 1e+113)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_1)) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / t_2);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * F), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-40], N[(N[Sqrt[N[(N[(C * 4.0), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+113], N[(N[(N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-40

   1. Initial program 15.4%

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

   3. Simplified 25.8%

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

   5. Taylor expanded in A around -inf 25.9%

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

      1. *-commutative 25.9%

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

   7. Simplified 25.9%

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

   if 1.9999999999999999e-40 < (pow.f64 B #s(literal 2 binary64)) < 1e113

   1. Initial program 39.3%

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

   3. Simplified 43.3%

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

      1. associate-*r* 43.3%

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

      2. associate-+r+ 42.7%

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

      3. hypot-undefine 39.3%

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

      4. unpow2 39.3%

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

      5. unpow2 39.3%

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

      6. +-commutative 39.3%

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

      7. sqrt-prod 39.3%

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

      8. *-commutative 39.3%

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

      9. associate-+l+ 39.8%

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

   6. Applied egg-rr 63.1%

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

   if 1e113 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.9%

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

   3. Taylor expanded in A around 0 8.1%

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

      1. mul-1-neg 8.1%

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

      2. *-commutative 8.1%

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

      3. *-commutative 8.1%

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

      4. +-commutative 8.1%

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

      5. unpow2 8.1%

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

      6. unpow2 8.1%

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

      7. hypot-define 23.0%

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

   5. Simplified 23.0%

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

      1. sqrt-prod 34.7%

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

   7. Applied egg-rr 34.7%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot 4\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+113}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot 4\right) \cdot \left(t\_0 \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 6e-37)
     (/ (sqrt (* (* C 4.0) (* t_0 F))) (- t_0))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 6e-37) {
		tmp = sqrt(((C * 4.0) * (t_0 * F))) / -t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 6e-37)
		tmp = Float64(sqrt(Float64(Float64(C * 4.0) * Float64(t_0 * F))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 6e-37], N[(N[Sqrt[N[(N[(C * 4.0), $MachinePrecision] * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 6e-37

   1. Initial program 15.1%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in A around -inf 25.5%

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

      1. *-commutative 25.5%

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

   7. Simplified 25.5%

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

   if 6e-37 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.4%

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

   3. Taylor expanded in A around 0 10.5%

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

      1. mul-1-neg 10.5%

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

      2. *-commutative 10.5%

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

      3. *-commutative 10.5%

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

      4. +-commutative 10.5%

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

      5. unpow2 10.5%

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

      6. unpow2 10.5%

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

      7. hypot-define 22.6%

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

   5. Simplified 22.6%

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

      1. sqrt-prod 32.1%

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

   7. Applied egg-rr 32.1%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot 4\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot F}\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 9 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot 4\right) \cdot \left(t\_1 \cdot F\right)}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 5.2 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \frac{t\_0}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 F))) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 9e-19)
     (/ (sqrt (* (* C 4.0) (* t_1 F))) (- t_1))
     (if (<= B_m 5.2e+200)
       (* (sqrt (+ C (hypot C B_m))) (/ t_0 (- B_m)))
       (/ t_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) {
	double t_0 = sqrt((2.0 * F));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 9e-19) {
		tmp = sqrt(((C * 4.0) * (t_1 * F))) / -t_1;
	} else if (B_m <= 5.2e+200) {
		tmp = sqrt((C + hypot(C, B_m))) * (t_0 / -B_m);
	} else {
		tmp = t_0 / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(2.0 * F))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 9e-19)
		tmp = Float64(sqrt(Float64(Float64(C * 4.0) * Float64(t_1 * F))) / Float64(-t_1));
	elseif (B_m <= 5.2e+200)
		tmp = Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(t_0 / Float64(-B_m)));
	else
		tmp = Float64(t_0 / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 9e-19], N[(N[Sqrt[N[(N[(C * 4.0), $MachinePrecision] * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[B$95$m, 5.2e+200], N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.00000000000000026e-19

   1. Initial program 16.3%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in A around -inf 17.1%

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

      1. *-commutative 17.1%

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

   7. Simplified 17.1%

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

   if 9.00000000000000026e-19 < B < 5.2000000000000003e200

   1. Initial program 18.8%

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

   3. Taylor expanded in A around 0 28.3%

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

      1. mul-1-neg 28.3%

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

      2. *-commutative 28.3%

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

      3. *-commutative 28.3%

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

      4. +-commutative 28.3%

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

      5. unpow2 28.3%

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

      6. unpow2 28.3%

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

      7. hypot-define 38.0%

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

   5. Simplified 38.0%

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

      1. add-exp-log 36.1%

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

      2. associate-*r/ 36.0%

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

      3. pow1/2 36.0%

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

      4. pow1/2 36.0%

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

      5. pow-prod-down 36.1%

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

   7. Applied egg-rr 36.1%

      \[\leadsto -\color{blue}{e^{\log \left(\frac{{\left(\left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot F\right) \cdot 2\right)}^{0.5}}{B}\right)}} \]
   8. Step-by-step derivation

      1. rem-exp-log 38.0%

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

      2. unpow-prod-down 37.7%

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

      3. pow1/2 37.7%

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

      4. sqrt-unprod 49.1%

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

      5. pow1/2 49.1%

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

      6. associate-*r/ 49.2%

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

      7. associate-*l* 49.1%

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

   9. Applied egg-rr 49.1%

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

       1. neg-sub0 49.1%

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

       2. associate-*r/ 49.2%

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

       3. pow1/2 49.2%

          \[\leadsto 0 - \sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{\color{blue}{{F}^{0.5}} \cdot \sqrt{2}}{B} \]

       4. pow1/2 49.2%

          \[\leadsto 0 - \sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{{F}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

       5. pow-prod-down 49.2%

          \[\leadsto 0 - \sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{\color{blue}{{\left(F \cdot 2\right)}^{0.5}}}{B} \]

   11. Applied egg-rr 49.2%

       \[\leadsto \color{blue}{0 - \sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{{\left(F \cdot 2\right)}^{0.5}}{B}} \]
   12. Step-by-step derivation

       1. sub0-neg 49.2%

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

       2. distribute-rgt-neg-out 49.2%

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

       3. distribute-neg-frac2 49.2%

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

       4. unpow1/2 49.2%

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

   13. Simplified 49.2%

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

   if 5.2000000000000003e200 < 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. Add Preprocessing

   3. Taylor expanded in B around inf 58.1%

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

      1. mul-1-neg 58.1%

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

      2. *-commutative 58.1%

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

   5. Simplified 58.1%

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

      1. *-commutative 58.1%

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

      2. pow1/2 58.1%

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

      3. pow1/2 58.1%

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

      4. pow-prod-down 58.4%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1/2 58.4%

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

   9. Simplified 58.4%

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

       1. metadata-eval 58.4%

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

       2. metadata-eval 58.4%

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

       3. sqrt-pow2 57.9%

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

       4. associate-*l/ 57.9%

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

       5. sqrt-div 86.2%

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

       6. sqrt-pow2 87.0%

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

       7. metadata-eval 87.0%

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

       8. metadata-eval 87.0%

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

   11. Applied egg-rr 87.0%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot 4\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{\sqrt{2 \cdot F}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot F}\\ \mathbf{if}\;B\_m \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{elif}\;B\_m \leq 9.4 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \frac{t\_0}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 F))))
   (if (<= B_m 5.8e-13)
     (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
     (if (<= B_m 9.4e+200)
       (* (sqrt (+ C (hypot C B_m))) (/ t_0 (- B_m)))
       (/ t_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) {
	double t_0 = sqrt((2.0 * F));
	double tmp;
	if (B_m <= 5.8e-13) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else if (B_m <= 9.4e+200) {
		tmp = sqrt((C + hypot(C, B_m))) * (t_0 / -B_m);
	} else {
		tmp = t_0 / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt((2.0 * F));
	double tmp;
	if (B_m <= 5.8e-13) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else if (B_m <= 9.4e+200) {
		tmp = Math.sqrt((C + Math.hypot(C, B_m))) * (t_0 / -B_m);
	} else {
		tmp = t_0 / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.sqrt((2.0 * F))
	tmp = 0
	if B_m <= 5.8e-13:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	elif B_m <= 9.4e+200:
		tmp = math.sqrt((C + math.hypot(C, B_m))) * (t_0 / -B_m)
	else:
		tmp = t_0 / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(2.0 * F))
	tmp = 0.0
	if (B_m <= 5.8e-13)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	elseif (B_m <= 9.4e+200)
		tmp = Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(t_0 / Float64(-B_m)));
	else
		tmp = Float64(t_0 / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt((2.0 * F));
	tmp = 0.0;
	if (B_m <= 5.8e-13)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	elseif (B_m <= 9.4e+200)
		tmp = sqrt((C + hypot(C, B_m))) * (t_0 / -B_m);
	else
		tmp = t_0 / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 5.8e-13], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 9.4e+200], N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.7999999999999995e-13

   1. Initial program 16.1%

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

   3. Taylor expanded in F around 0 17.5%

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

   5. Simplified 24.5%

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

   6. Taylor expanded in C around inf 14.5%

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

   if 5.7999999999999995e-13 < B < 9.3999999999999995e200

   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. Add Preprocessing

   3. Taylor expanded in A around 0 29.9%

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

      1. mul-1-neg 29.9%

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

      2. *-commutative 29.9%

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

      3. *-commutative 29.9%

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

      4. +-commutative 29.9%

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

      5. unpow2 29.9%

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

      6. unpow2 29.9%

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

      7. hypot-define 40.1%

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

   5. Simplified 40.1%

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

      1. add-exp-log 38.1%

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

      2. associate-*r/ 38.1%

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

      3. pow1/2 38.1%

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

      4. pow1/2 38.1%

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

      5. pow-prod-down 38.1%

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

   7. Applied egg-rr 38.1%

      \[\leadsto -\color{blue}{e^{\log \left(\frac{{\left(\left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot F\right) \cdot 2\right)}^{0.5}}{B}\right)}} \]
   8. Step-by-step derivation

      1. rem-exp-log 40.2%

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

      2. unpow-prod-down 39.9%

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

      3. pow1/2 39.9%

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

      4. sqrt-unprod 52.0%

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

      5. pow1/2 52.0%

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

      6. associate-*r/ 52.0%

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

      7. associate-*l* 51.9%

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

   9. Applied egg-rr 51.9%

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

       1. neg-sub0 51.9%

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

       2. associate-*r/ 52.1%

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

       3. pow1/2 52.1%

          \[\leadsto 0 - \sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{\color{blue}{{F}^{0.5}} \cdot \sqrt{2}}{B} \]

       4. pow1/2 52.1%

          \[\leadsto 0 - \sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{{F}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

       5. pow-prod-down 52.1%

          \[\leadsto 0 - \sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{\color{blue}{{\left(F \cdot 2\right)}^{0.5}}}{B} \]

   11. Applied egg-rr 52.1%

       \[\leadsto \color{blue}{0 - \sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{{\left(F \cdot 2\right)}^{0.5}}{B}} \]
   12. Step-by-step derivation

       1. sub0-neg 52.1%

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

       2. distribute-rgt-neg-out 52.1%

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

       3. distribute-neg-frac2 52.1%

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

       4. unpow1/2 52.1%

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

   13. Simplified 52.1%

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

   if 9.3999999999999995e200 < 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. Add Preprocessing

   3. Taylor expanded in B around inf 58.1%

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

      1. mul-1-neg 58.1%

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

      2. *-commutative 58.1%

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

   5. Simplified 58.1%

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

      1. *-commutative 58.1%

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

      2. pow1/2 58.1%

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

      3. pow1/2 58.1%

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

      4. pow-prod-down 58.4%

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

   7. Applied egg-rr 58.4%

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

      1. unpow1/2 58.4%

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

   9. Simplified 58.4%

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

       1. metadata-eval 58.4%

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

       2. metadata-eval 58.4%

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

       3. sqrt-pow2 57.9%

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

       4. associate-*l/ 57.9%

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

       5. sqrt-div 86.2%

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

       6. sqrt-pow2 87.0%

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

       7. metadata-eval 87.0%

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

       8. metadata-eval 87.0%

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

   11. Applied egg-rr 87.0%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{elif}\;B \leq 9.4 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \frac{\sqrt{2 \cdot F}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{elif}\;B\_m \leq 9.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6e-13)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
   (if (<= B_m 9.5e+124)
     (/ (sqrt (* 2.0 (* F (+ C (hypot C B_m))))) (- B_m))
     (/ (sqrt (* 2.0 F)) (- (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) {
	double tmp;
	if (B_m <= 6e-13) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else if (B_m <= 9.5e+124) {
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6e-13) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else if (B_m <= 9.5e+124) {
		tmp = Math.sqrt((2.0 * (F * (C + Math.hypot(C, B_m))))) / -B_m;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 6e-13:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	elif B_m <= 9.5e+124:
		tmp = math.sqrt((2.0 * (F * (C + math.hypot(C, B_m))))) / -B_m
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6e-13)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	elseif (B_m <= 9.5e+124)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m))))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 6e-13)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	elseif (B_m <= 9.5e+124)
		tmp = sqrt((2.0 * (F * (C + hypot(C, B_m))))) / -B_m;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6e-13], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 9.5e+124], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.99999999999999968e-13

   1. Initial program 16.1%

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

   3. Taylor expanded in F around 0 17.5%

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

   5. Simplified 24.5%

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

   6. Taylor expanded in C around inf 14.5%

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

   if 5.99999999999999968e-13 < B < 9.50000000000000004e124

   1. Initial program 32.4%

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

   3. Taylor expanded in A around 0 44.1%

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

      1. mul-1-neg 44.1%

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

      2. *-commutative 44.1%

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

      3. *-commutative 44.1%

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

      4. +-commutative 44.1%

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

      5. unpow2 44.1%

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

      6. unpow2 44.1%

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

      7. hypot-define 44.6%

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

   5. Simplified 44.6%

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

      1. neg-sub0 44.6%

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

      2. associate-*r/ 44.4%

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

      3. pow1/2 44.4%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot F\right)}^{0.5}} \cdot \sqrt{2}}{B} \]

      4. pow1/2 44.4%

         \[\leadsto 0 - \frac{{\left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot F\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}}{B} \]

      5. pow-prod-down 44.8%

         \[\leadsto 0 - \frac{\color{blue}{{\left(\left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot F\right) \cdot 2\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 44.8%

      \[\leadsto \color{blue}{0 - \frac{{\left(\left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot F\right) \cdot 2\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 44.8%

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

      2. distribute-neg-frac2 44.8%

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

      3. unpow1/2 44.8%

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

   9. Simplified 44.8%

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

   if 9.50000000000000004e124 < B

   1. Initial program 0.3%

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

   3. Taylor expanded in B around inf 52.5%

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

      1. mul-1-neg 52.5%

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

      2. *-commutative 52.5%

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

   5. Simplified 52.5%

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

      1. *-commutative 52.5%

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

      2. pow1/2 52.5%

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

      3. pow1/2 52.5%

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

      4. pow-prod-down 52.7%

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

   7. Applied egg-rr 52.7%

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

      1. unpow1/2 52.7%

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

   9. Simplified 52.7%

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

       1. metadata-eval 52.7%

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

       2. metadata-eval 52.7%

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

       3. sqrt-pow2 52.4%

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

       4. associate-*l/ 52.3%

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

       5. sqrt-div 71.0%

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

       6. sqrt-pow2 71.5%

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

       7. metadata-eval 71.5%

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

       8. metadata-eval 71.5%

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

   11. Applied egg-rr 71.5%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6.5e-13)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
   (/ (sqrt (* 2.0 F)) (- (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) {
	double tmp;
	if (B_m <= 6.5e-13) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 6.5d-13) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / a)))
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6.5e-13) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 6.5e-13:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6.5e-13)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 6.5e-13)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6.5e-13], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.49999999999999957e-13

   1. Initial program 16.1%

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

   3. Taylor expanded in F around 0 17.5%

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

   5. Simplified 24.5%

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

   6. Taylor expanded in C around inf 14.5%

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

   if 6.49999999999999957e-13 < B

   1. Initial program 13.0%

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

   3. Taylor expanded in B around inf 45.9%

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

      1. mul-1-neg 45.9%

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

      2. *-commutative 45.9%

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

   5. Simplified 45.9%

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

      1. *-commutative 45.9%

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

      2. pow1/2 45.9%

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

      3. pow1/2 45.9%

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

      4. pow-prod-down 46.1%

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

   7. Applied egg-rr 46.1%

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

      1. unpow1/2 46.1%

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

   9. Simplified 46.1%

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

       1. metadata-eval 46.1%

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

       2. metadata-eval 46.1%

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

       3. sqrt-pow2 45.8%

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

       4. associate-*l/ 45.7%

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

       5. sqrt-div 58.4%

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

       6. sqrt-pow2 58.8%

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

       7. metadata-eval 58.8%

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

       8. metadata-eval 58.8%

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

   11. Applied egg-rr 58.8%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.5% accurate, 3.1× speedup

Math

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

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 (* 2.0 F)) (- (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((2.0 * F)) / -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((2.0d0 * f)) / -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((2.0 * F)) / -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((2.0 * F)) / -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(2.0 * F)) / 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((2.0 * F)) / -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[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 15.3%

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

3. Taylor expanded in B around inf 12.9%

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

   1. mul-1-neg 12.9%

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

   2. *-commutative 12.9%

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

5. Simplified 12.9%

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

   1. *-commutative 12.9%

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

   2. pow1/2 13.1%

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

   3. pow1/2 13.1%

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

   4. pow-prod-down 13.1%

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

7. Applied egg-rr 13.1%

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

   1. unpow1/2 12.9%

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

9. Simplified 12.9%

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

    1. metadata-eval 12.9%

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

    2. metadata-eval 12.9%

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

    3. sqrt-pow2 12.9%

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

    4. associate-*l/ 12.8%

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

    5. sqrt-div 15.9%

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

    6. sqrt-pow2 16.0%

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

    7. metadata-eval 16.0%

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

    8. metadata-eval 16.0%

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

11. Applied egg-rr 16.0%

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

12. Final simplification 16.0%

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

Alternative 8: 35.6% accurate, 3.1× speedup

Math

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

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 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(F);
}

Fortran

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

Java

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

Derivation

1. Initial program 15.3%

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

3. Taylor expanded in B around inf 12.9%

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

   1. mul-1-neg 12.9%

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

   2. *-commutative 12.9%

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

5. Simplified 12.9%

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

   1. *-commutative 12.9%

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

   2. pow1/2 13.1%

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

   3. pow1/2 13.1%

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

   4. pow-prod-down 13.1%

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

7. Applied egg-rr 13.1%

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

   1. unpow1/2 12.9%

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

9. Simplified 12.9%

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

    1. neg-sub0 12.9%

       \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]

11. Applied egg-rr 12.9%

    \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
12. Step-by-step derivation

    1. neg-sub0 12.9%

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

    2. associate-*l/ 12.9%

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

    3. associate-/l* 12.9%

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

13. Simplified 12.9%

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

    1. *-commutative 12.9%

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

    2. sqrt-prod 15.9%

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

15. Applied egg-rr 15.9%

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

16. Final simplification 15.9%

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

Alternative 9: 27.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.3%

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

3. Taylor expanded in B around inf 12.9%

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

   1. mul-1-neg 12.9%

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

   2. *-commutative 12.9%

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

5. Simplified 12.9%

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

   1. *-commutative 12.9%

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

   2. pow1/2 13.1%

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

   3. pow1/2 13.1%

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

   4. pow-prod-down 13.1%

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

7. Applied egg-rr 13.1%

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

   1. unpow1/2 12.9%

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

9. Simplified 12.9%

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

    1. neg-sub0 12.9%

       \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]

11. Applied egg-rr 12.9%

    \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
12. Step-by-step derivation

    1. neg-sub0 12.9%

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

    2. associate-*l/ 12.9%

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

    3. associate-/l* 12.9%

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

13. Simplified 12.9%

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

    1. add-sqr-sqrt 12.9%

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

    2. pow1/2 12.9%

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

    3. pow1/2 13.1%

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

    4. pow-prod-down 18.2%

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

    5. pow2 18.2%

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

15. Applied egg-rr 18.2%

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

    1. unpow1/2 18.2%

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

    2. unpow2 18.2%

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

    3. rem-sqrt-square 28.5%

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

    4. associate-*r/ 28.5%

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

17. Simplified 28.5%

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

18. Final simplification 28.5%

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

Alternative 10: 27.4% 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 15.3%

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

3. Taylor expanded in B around inf 12.9%

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

   1. mul-1-neg 12.9%

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

   2. *-commutative 12.9%

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

5. Simplified 12.9%

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

   1. *-commutative 12.9%

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

   2. pow1/2 13.1%

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

   3. pow1/2 13.1%

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

   4. pow-prod-down 13.1%

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

7. Applied egg-rr 13.1%

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

8. Final simplification 13.1%

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

Alternative 11: 27.3% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.3%

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

3. Taylor expanded in B around inf 12.9%

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

   1. mul-1-neg 12.9%

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

   2. *-commutative 12.9%

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

5. Simplified 12.9%

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

   1. *-commutative 12.9%

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

   2. pow1/2 13.1%

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

   3. pow1/2 13.1%

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

   4. pow-prod-down 13.1%

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

7. Applied egg-rr 13.1%

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

   1. unpow1/2 12.9%

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

9. Simplified 12.9%

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

    1. neg-sub0 12.9%

       \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]

11. Applied egg-rr 12.9%

    \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B} \cdot 2}} \]
12. Step-by-step derivation

    1. neg-sub0 12.9%

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

    2. associate-*l/ 12.9%

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

    3. associate-/l* 12.9%

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

13. Simplified 12.9%

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

Reproduce

herbie shell --seed 2024145 
(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))))