ABCF->ab-angle a

Percentage Accurate: 18.6% → 56.3%

Time: 14.8s

Alternatives: 8

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(t\_0 \cdot 2\right)\right) \cdot \left(C \cdot 2\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 6.7 \cdot 10^{+227}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{B\_m} \cdot \left(-\sqrt{\mathsf{hypot}\left(A - C, B\_m\right) + \left(A + C\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

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 (* A C) -4.0 (* B_m B_m))))
   (if (<= B_m 3.45e-9)
     (/ (sqrt (* (* F (* t_0 2.0)) (* C 2.0))) (- t_0))
     (if (<= B_m 6.7e+227)
       (* (/ (sqrt (* F 2.0)) B_m) (- (sqrt (+ (hypot (- A C) B_m) (+ A C)))))
       (* (- (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((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 3.45e-9) {
		tmp = sqrt(((F * (t_0 * 2.0)) * (C * 2.0))) / -t_0;
	} else if (B_m <= 6.7e+227) {
		tmp = (sqrt((F * 2.0)) / B_m) * -sqrt((hypot((A - C), B_m) + (A + C)));
	} else {
		tmp = -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(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.45e-9)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(t_0 * 2.0)) * Float64(C * 2.0))) / Float64(-t_0));
	elseif (B_m <= 6.7e+227)
		tmp = Float64(Float64(sqrt(Float64(F * 2.0)) / B_m) * Float64(-sqrt(Float64(hypot(Float64(A - C), B_m) + Float64(A + C)))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / 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[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.45e-9], N[(N[Sqrt[N[(N[(F * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 6.7e+227], N[(N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.44999999999999987e-9

   1. Initial program 18.4%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in C around inf

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

   8. Applied rewrites 16.4%

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

   10. Applied rewrites 17.3%

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

   if 3.44999999999999987e-9 < B < 6.7000000000000002e227

   1. Initial program 37.1%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 71.4

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

   7. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lower-neg.f64 71.4

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 71.7

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

   9. Applied rewrites 71.7%

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

   if 6.7000000000000002e227 < 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

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 32.4

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

   5. Applied rewrites 32.4%

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

   7. Applied rewrites 32.6%

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

   9. Applied rewrites 78.0%

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

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right)\right) \cdot \left(C \cdot 2\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6.7 \cdot 10^{+227}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{B} \cdot \left(-\sqrt{\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(t\_0 \cdot 2\right)\right) \cdot \left(C \cdot 2\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 6.7 \cdot 10^{+227}:\\ \;\;\;\;\left(-\sqrt{\mathsf{hypot}\left(B\_m, C\right) + C}\right) \cdot \frac{\sqrt{F \cdot 2}}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

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 (* A C) -4.0 (* B_m B_m))))
   (if (<= B_m 3.6e-9)
     (/ (sqrt (* (* F (* t_0 2.0)) (* C 2.0))) (- t_0))
     (if (<= B_m 6.7e+227)
       (* (- (sqrt (+ (hypot B_m C) C))) (/ (sqrt (* F 2.0)) 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((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 3.6e-9) {
		tmp = sqrt(((F * (t_0 * 2.0)) * (C * 2.0))) / -t_0;
	} else if (B_m <= 6.7e+227) {
		tmp = -sqrt((hypot(B_m, C) + C)) * (sqrt((F * 2.0)) / B_m);
	} else {
		tmp = -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(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.6e-9)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(t_0 * 2.0)) * Float64(C * 2.0))) / Float64(-t_0));
	elseif (B_m <= 6.7e+227)
		tmp = Float64(Float64(-sqrt(Float64(hypot(B_m, C) + C))) * Float64(sqrt(Float64(F * 2.0)) / B_m));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / 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[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.6e-9], N[(N[Sqrt[N[(N[(F * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 6.7e+227], N[((-N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.6e-9

   1. Initial program 18.4%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in C around inf

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

   8. Applied rewrites 16.4%

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

   10. Applied rewrites 17.3%

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

   if 3.6e-9 < B < 6.7000000000000002e227

   1. Initial program 37.1%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 71.4

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

   7. Applied rewrites 71.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-rgt-identity N/A

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

      5. lower-neg.f64 71.4

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 71.7

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

   9. Applied rewrites 71.7%

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

   10. Taylor expanded in A around 0

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

       1. lower-+.f64 N/A

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

       2. unpow2 N/A

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

       3. unpow2 N/A

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

       4. lower-hypot.f64 60.1

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

   12. Applied rewrites 60.1%

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

   if 6.7000000000000002e227 < 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

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 32.4

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

   5. Applied rewrites 32.4%

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

   7. Applied rewrites 32.6%

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

   9. Applied rewrites 78.0%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right)\right) \cdot \left(C \cdot 2\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6.7 \cdot 10^{+227}:\\ \;\;\;\;\left(-\sqrt{\mathsf{hypot}\left(B, C\right) + C}\right) \cdot \frac{\sqrt{F \cdot 2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(t\_0 \cdot 2\right)\right) \cdot \left(C \cdot 2\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 6 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

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 (* A C) -4.0 (* B_m B_m))))
   (if (<= B_m 1.6e-6)
     (/ (sqrt (* (* F (* t_0 2.0)) (* C 2.0))) (- t_0))
     (if (<= B_m 6e+122)
       (* (sqrt (* (+ (hypot C B_m) C) F)) (/ (sqrt 2.0) (- 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((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 1.6e-6) {
		tmp = sqrt(((F * (t_0 * 2.0)) * (C * 2.0))) / -t_0;
	} else if (B_m <= 6e+122) {
		tmp = sqrt(((hypot(C, B_m) + C) * F)) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -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(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 1.6e-6)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(t_0 * 2.0)) * Float64(C * 2.0))) / Float64(-t_0));
	elseif (B_m <= 6e+122)
		tmp = Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / 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[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.6e-6], N[(N[Sqrt[N[(N[(F * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 6e+122], N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.5999999999999999e-6

   1. Initial program 18.6%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 23.0%

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

   6. Taylor expanded in C around inf

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

   8. Applied rewrites 16.7%

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

   10. Applied rewrites 17.6%

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

   if 1.5999999999999999e-6 < B < 5.99999999999999972e122

   1. Initial program 45.5%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. unpow2 N/A

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

      14. unpow2 N/A

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

      15. lower-hypot.f64 58.7

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

   5. Applied rewrites 58.7%

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

   if 5.99999999999999972e122 < B

   1. Initial program 10.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 39.7

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

   5. Applied rewrites 39.7%

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

   7. Applied rewrites 39.9%

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

   9. Applied rewrites 69.9%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right)\right) \cdot \left(C \cdot 2\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(t\_0 \cdot 2\right)\right) \cdot \left(C \cdot 2\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

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 (* A C) -4.0 (* B_m B_m))))
   (if (<= B_m 3.8e-9)
     (/ (sqrt (* (* F (* t_0 2.0)) (* C 2.0))) (- t_0))
     (* (- (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((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 3.8e-9) {
		tmp = sqrt(((F * (t_0 * 2.0)) * (C * 2.0))) / -t_0;
	} else {
		tmp = -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(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.8e-9)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(t_0 * 2.0)) * Float64(C * 2.0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / 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[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.8e-9], N[(N[Sqrt[N[(N[(F * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.80000000000000011e-9

   1. Initial program 18.4%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in C around inf

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

   8. Applied rewrites 16.4%

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

   10. Applied rewrites 17.3%

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

   if 3.80000000000000011e-9 < B

   1. Initial program 27.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 44.8

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

   5. Applied rewrites 44.8%

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

   7. Applied rewrites 45.0%

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

   9. Applied rewrites 60.0%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right) \cdot 2\right)\right) \cdot \left(C \cdot 2\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9.8 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\frac{F}{A} \cdot -0.5} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

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
 (if (<= B_m 9.8e-32)
   (* (sqrt (* (/ F A) -0.5)) (- (sqrt 2.0)))
   (* (- (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 tmp;
	if (B_m <= 9.8e-32) {
		tmp = sqrt(((F / A) * -0.5)) * -sqrt(2.0);
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 9.8d-32) then
        tmp = sqrt(((f / a) * (-0.5d0))) * -sqrt(2.0d0)
    else
        tmp = -sqrt(f) * sqrt((2.0d0 / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.8e-32) {
		tmp = Math.sqrt(((F / A) * -0.5)) * -Math.sqrt(2.0);
	} else {
		tmp = -Math.sqrt(F) * Math.sqrt((2.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9.8e-32:
		tmp = math.sqrt(((F / A) * -0.5)) * -math.sqrt(2.0)
	else:
		tmp = -math.sqrt(F) * math.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)
	tmp = 0.0
	if (B_m <= 9.8e-32)
		tmp = Float64(sqrt(Float64(Float64(F / A) * -0.5)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 9.8e-32)
		tmp = sqrt(((F / A) * -0.5)) * -sqrt(2.0);
	else
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9.8e-32], N[(N[Sqrt[N[(N[(F / A), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9.7999999999999996e-32

   1. Initial program 18.3%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 19.6%

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

   6. Taylor expanded in C around inf

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

   8. Applied rewrites 15.9%

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

   if 9.7999999999999996e-32 < B

   1. Initial program 27.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 41.6

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

   5. Applied rewrites 41.6%

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

   7. Applied rewrites 41.8%

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

   9. Applied rewrites 55.6%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.8 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{\frac{F}{A} \cdot -0.5} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.6% accurate, 12.6× speedup

Math

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

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)) (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) {
	return -sqrt(F) * sqrt((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) * sqrt((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) * Math.sqrt((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) * math.sqrt((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(Float64(-sqrt(F)) * sqrt(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) * sqrt((2.0 / B_m));
end

Wolfram

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

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 11.9

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

5. Applied rewrites 11.9%

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

7. Applied rewrites 11.9%

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

9. Applied rewrites 14.8%

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

10. Final simplification 14.8%

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

Alternative 7: 27.5% accurate, 16.9× 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 20.5%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 11.9

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

5. Applied rewrites 11.9%

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

7. Applied rewrites 11.9%

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

Alternative 8: 27.5% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 11.9

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

5. Applied rewrites 11.9%

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

7. Applied rewrites 11.9%

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

9. Applied rewrites 11.9%

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

Reproduce

herbie shell --seed 2024284 
(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))))