ABCF->ab-angle b

Percentage Accurate: 18.9% → 49.5%

Time: 14.3s

Alternatives: 11

Speedup: 18.9×

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: 18.9% 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: 49.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{-F}\\ t_1 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_3 := C \cdot \left(A \cdot 4\right) - {B}^{2}\\ t_4 := \frac{\sqrt{\left(\left(t\_3 \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(-{C}^{-0.5}\right) \cdot t\_0\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{\left(t\_2 \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right)}}{-t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{-1}{t\_1} \cdot \sqrt{\left(\left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right) \cdot \left(F \cdot 2\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{C}}{t\_0}}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (- F)))
        (t_1 (fma (* -4.0 C) A (* B B)))
        (t_2 (fma -4.0 (* C A) (* B B)))
        (t_3 (- (* C (* A 4.0)) (pow B 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (* (* t_3 F) 2.0)
            (- (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))) (+ C A))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (* (- (pow C -0.5)) t_0)
     (if (<= t_4 -5e-187)
       (/ (sqrt (* (* t_2 (* F 2.0)) (- (+ C A) (hypot (- A C) B)))) (- t_2))
       (if (<= t_4 INFINITY)
         (*
          (/ -1.0 t_1)
          (sqrt (* (* (+ (fma (/ (* B B) C) -0.5 A) A) (* F 2.0)) t_1)))
         (/ -1.0 (/ (sqrt C) t_0)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(-F);
	double t_1 = fma((-4.0 * C), A, (B * B));
	double t_2 = fma(-4.0, (C * A), (B * B));
	double t_3 = (C * (A * 4.0)) - pow(B, 2.0);
	double t_4 = sqrt((((t_3 * F) * 2.0) * (sqrt((pow((A - C), 2.0) + pow(B, 2.0))) - (C + A)))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -pow(C, -0.5) * t_0;
	} else if (t_4 <= -5e-187) {
		tmp = sqrt(((t_2 * (F * 2.0)) * ((C + A) - hypot((A - C), B)))) / -t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (-1.0 / t_1) * sqrt((((fma(((B * B) / C), -0.5, A) + A) * (F * 2.0)) * t_1));
	} else {
		tmp = -1.0 / (sqrt(C) / t_0);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = sqrt(Float64(-F))
	t_1 = fma(Float64(-4.0 * C), A, Float64(B * B))
	t_2 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_3 = Float64(Float64(C * Float64(A * 4.0)) - (B ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(Float64(t_3 * F) * 2.0) * Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))) - Float64(C + A)))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(-(C ^ -0.5)) * t_0);
	elseif (t_4 <= -5e-187)
		tmp = Float64(sqrt(Float64(Float64(t_2 * Float64(F * 2.0)) * Float64(Float64(C + A) - hypot(Float64(A - C), B)))) / Float64(-t_2));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(-1.0 / t_1) * sqrt(Float64(Float64(Float64(fma(Float64(Float64(B * B) / C), -0.5, A) + A) * Float64(F * 2.0)) * t_1)));
	else
		tmp = Float64(-1.0 / Float64(sqrt(C) / t_0));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[(-F)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision] - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(N[(t$95$3 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[((-N[Power[C, -0.5], $MachinePrecision]) * t$95$0), $MachinePrecision], If[LessEqual[t$95$4, -5e-187], N[(N[Sqrt[N[(N[(t$95$2 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(-1.0 / t$95$1), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[C], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.1%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 32.6%

      \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
   8. Step-by-step derivation

   9. Applied rewrites 32.7%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999996e-187

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

   if -4.9999999999999996e-187 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 19.4%

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

   3. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 33.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) - \color{blue}{\left(-A\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 33.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) - \left(-A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 33.5%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   9. Applied rewrites 36.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 14.1%

      \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
   8. Step-by-step derivation

   9. Applied rewrites 14.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{C}}{-\sqrt{-F}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(-{C}^{-0.5}\right) \cdot \sqrt{-F}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{\left(\left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right) \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{C}}{\sqrt{-F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 36.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(t\_0 \cdot \left(F \cdot 2\right)\right) \cdot \left(A + A\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\left(-{C}^{-0.5}\right) \cdot \sqrt{-F}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 C) A (* B B))))
   (if (<= (pow B 2.0) 5e-290)
     (* (sqrt (* (* t_0 (* F 2.0)) (+ A A))) (/ -1.0 t_0))
     (if (<= (pow B 2.0) 2e+82)
       (* (- (pow C -0.5)) (sqrt (- F)))
       (* (/ (- (sqrt 2.0)) B) (sqrt (* (- A (hypot A B)) F)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma((-4.0 * C), A, (B * B));
	double tmp;
	if (pow(B, 2.0) <= 5e-290) {
		tmp = sqrt(((t_0 * (F * 2.0)) * (A + A))) * (-1.0 / t_0);
	} else if (pow(B, 2.0) <= 2e+82) {
		tmp = -pow(C, -0.5) * sqrt(-F);
	} else {
		tmp = (-sqrt(2.0) / B) * sqrt(((A - hypot(A, B)) * F));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(Float64(-4.0 * C), A, Float64(B * B))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e-290)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(F * 2.0)) * Float64(A + A))) * Float64(-1.0 / t_0));
	elseif ((B ^ 2.0) <= 2e+82)
		tmp = Float64(Float64(-(C ^ -0.5)) * sqrt(Float64(-F)));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(Float64(A - hypot(A, B)) * F)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-290], N[(N[Sqrt[N[(N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e+82], N[((-N[Power[C, -0.5], $MachinePrecision]) * N[Sqrt[(-F)], $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[Sqrt[N[(N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000001e-290

   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 C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 26.7

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

   5. Applied rewrites 26.7%

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

   7. Applied rewrites 26.7%

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

   if 5.0000000000000001e-290 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e82

   1. Initial program 26.8%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 27.4%

      \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
   8. Step-by-step derivation

   9. Applied rewrites 27.4%

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

   if 1.9999999999999999e82 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.9%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 23.8

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

   5. Applied rewrites 23.8%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + A\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+82}:\\ \;\;\;\;\left(-{C}^{-0.5}\right) \cdot \sqrt{-F}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 36.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+108}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{C}}\right) \cdot \sqrt{-F}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{\frac{C + A}{B} - 1}{B} \cdot \left(F \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 1e+108)
   (* (- (sqrt (/ 1.0 C))) (sqrt (- F)))
   (- (sqrt (* (/ (- (/ (+ C A) B) 1.0) B) (* F 2.0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 1e+108) {
		tmp = -sqrt((1.0 / C)) * sqrt(-F);
	} else {
		tmp = -sqrt((((((C + A) / B) - 1.0) / B) * (F * 2.0)));
	}
	return tmp;
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b ** 2.0d0) <= 1d+108) then
        tmp = -sqrt((1.0d0 / c)) * sqrt(-f)
    else
        tmp = -sqrt((((((c + a) / b) - 1.0d0) / b) * (f * 2.0d0)))
    end if
    code = tmp
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (Math.pow(B, 2.0) <= 1e+108) {
		tmp = -Math.sqrt((1.0 / C)) * Math.sqrt(-F);
	} else {
		tmp = -Math.sqrt((((((C + A) / B) - 1.0) / B) * (F * 2.0)));
	}
	return tmp;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	tmp = 0
	if math.pow(B, 2.0) <= 1e+108:
		tmp = -math.sqrt((1.0 / C)) * math.sqrt(-F)
	else:
		tmp = -math.sqrt((((((C + A) / B) - 1.0) / B) * (F * 2.0)))
	return tmp

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 1e+108)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / C))) * sqrt(Float64(-F)));
	else
		tmp = Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(C + A) / B) - 1.0) / B) * Float64(F * 2.0))));
	end
	return tmp
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if ((B ^ 2.0) <= 1e+108)
		tmp = -sqrt((1.0 / C)) * sqrt(-F);
	else
		tmp = -sqrt((((((C + A) / B) - 1.0) / B) * (F * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+108], N[((-N[Sqrt[N[(1.0 / C), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[(-F)], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(N[(N[(N[(C + A), $MachinePrecision] / B), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e108

   1. Initial program 23.4%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 23.7%

      \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.7%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 23.7%

       \[\leadsto \sqrt{-F} \cdot \left(-\sqrt{\frac{1}{C}}\right) \]

   if 1e108 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.8%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 26.1%

      \[\leadsto -\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \sqrt{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 26.2%

       \[\leadsto \color{blue}{-\sqrt{\frac{\frac{C + A}{B} - 1}{B} \cdot \left(F \cdot 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+108}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{C}}\right) \cdot \sqrt{-F}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{\frac{C + A}{B} - 1}{B} \cdot \left(F \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 5.2 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(-B, B, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-{C}^{-0.5}\right) \cdot \sqrt{-F}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= C 5.2e-247)
   (/
    (sqrt (* (* (fma (* -4.0 C) A (* B B)) (* F 2.0)) (+ A A)))
    (fma (- B) B (* C (* A 4.0))))
   (* (- (pow C -0.5)) (sqrt (- F)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 5.2e-247) {
		tmp = sqrt(((fma((-4.0 * C), A, (B * B)) * (F * 2.0)) * (A + A))) / fma(-B, B, (C * (A * 4.0)));
	} else {
		tmp = -pow(C, -0.5) * sqrt(-F);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (C <= 5.2e-247)
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(-4.0 * C), A, Float64(B * B)) * Float64(F * 2.0)) * Float64(A + A))) / fma(Float64(-B), B, Float64(C * Float64(A * 4.0))));
	else
		tmp = Float64(Float64(-(C ^ -0.5)) * sqrt(Float64(-F)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[C, 5.2e-247], N[(N[Sqrt[N[(N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-B) * B + N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Power[C, -0.5], $MachinePrecision]) * N[Sqrt[(-F)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 5.2e-247

   1. Initial program 24.3%

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

   3. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 9.5

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

   5. Applied rewrites 9.5%

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

   7. Applied rewrites 9.5%

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

   if 5.2e-247 < C

   1. Initial program 11.5%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 43.4%

      \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
   8. Step-by-step derivation

   9. Applied rewrites 43.4%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 5.2 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(-B, B, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-{C}^{-0.5}\right) \cdot \sqrt{-F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 5.2 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(-B, B, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{C}}\right) \cdot \sqrt{-F}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= C 5.2e-247)
   (/
    (sqrt (* (* (fma (* -4.0 C) A (* B B)) (* F 2.0)) (+ A A)))
    (fma (- B) B (* C (* A 4.0))))
   (* (- (sqrt (/ 1.0 C))) (sqrt (- F)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 5.2e-247) {
		tmp = sqrt(((fma((-4.0 * C), A, (B * B)) * (F * 2.0)) * (A + A))) / fma(-B, B, (C * (A * 4.0)));
	} else {
		tmp = -sqrt((1.0 / C)) * sqrt(-F);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (C <= 5.2e-247)
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(-4.0 * C), A, Float64(B * B)) * Float64(F * 2.0)) * Float64(A + A))) / fma(Float64(-B), B, Float64(C * Float64(A * 4.0))));
	else
		tmp = Float64(Float64(-sqrt(Float64(1.0 / C))) * sqrt(Float64(-F)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[C, 5.2e-247], N[(N[Sqrt[N[(N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[((-B) * B + N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(1.0 / C), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[(-F)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 5.2e-247

   1. Initial program 24.3%

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

   3. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 9.5

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

   5. Applied rewrites 9.5%

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

   7. Applied rewrites 9.5%

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

   if 5.2e-247 < C

   1. Initial program 11.5%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 43.4%

      \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
   8. Step-by-step derivation

   9. Applied rewrites 43.4%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 43.4%

       \[\leadsto \sqrt{-F} \cdot \left(-\sqrt{\frac{1}{C}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 5.2 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(-B, B, C \cdot \left(A \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{C}}\right) \cdot \sqrt{-F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -1.9 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 3.4 \cdot 10^{-200}:\\ \;\;\;\;-\sqrt{\frac{\frac{C + A}{B} - 1}{B} \cdot \left(F \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{C}}\right) \cdot \sqrt{-F}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= C -1.9e-130)
   (*
    (sqrt (* (* (* (* (+ A A) F) C) A) -8.0))
    (/ -1.0 (fma (* -4.0 C) A (* B B))))
   (if (<= C 3.4e-200)
     (- (sqrt (* (/ (- (/ (+ C A) B) 1.0) B) (* F 2.0))))
     (* (- (sqrt (/ 1.0 C))) (sqrt (- F))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -1.9e-130) {
		tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) * (-1.0 / fma((-4.0 * C), A, (B * B)));
	} else if (C <= 3.4e-200) {
		tmp = -sqrt((((((C + A) / B) - 1.0) / B) * (F * 2.0)));
	} else {
		tmp = -sqrt((1.0 / C)) * sqrt(-F);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (C <= -1.9e-130)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) * Float64(-1.0 / fma(Float64(-4.0 * C), A, Float64(B * B))));
	elseif (C <= 3.4e-200)
		tmp = Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(C + A) / B) - 1.0) / B) * Float64(F * 2.0))));
	else
		tmp = Float64(Float64(-sqrt(Float64(1.0 / C))) * sqrt(Float64(-F)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[C, -1.9e-130], N[(N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 3.4e-200], (-N[Sqrt[N[(N[(N[(N[(N[(C + A), $MachinePrecision] / B), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[((-N[Sqrt[N[(1.0 / C), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[(-F)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.8999999999999999e-130

   1. Initial program 23.2%

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

   3. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 7.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) - \color{blue}{\left(-A\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 7.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) - \left(-A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 7.9%

      \[\leadsto \color{blue}{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}} \]

   8. Taylor expanded in C around inf

      \[\leadsto \sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{-8 \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A - -1 \cdot A\right)\right)}\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

      7. lower-neg.f64 8.1

         \[\leadsto \sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

   10. Applied rewrites 8.1%

       \[\leadsto \sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \]

   if -1.8999999999999999e-130 < C < 3.4000000000000003e-200

   1. Initial program 26.4%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 19.8%

      \[\leadsto -\sqrt{F \cdot \frac{\frac{A + C}{B} - 1}{B}} \cdot \sqrt{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 19.9%

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

   if 3.4000000000000003e-200 < C

   1. Initial program 10.9%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 44.9%

      \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
   8. Step-by-step derivation

   9. Applied rewrites 44.9%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 44.9%

       \[\leadsto \sqrt{-F} \cdot \left(-\sqrt{\frac{1}{C}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.9 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8} \cdot \frac{-1}{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 3.4 \cdot 10^{-200}:\\ \;\;\;\;-\sqrt{\frac{\frac{C + A}{B} - 1}{B} \cdot \left(F \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{C}}\right) \cdot \sqrt{-F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.9% accurate, 9.8× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{C}}\right) \cdot \sqrt{-F}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{-1}{B} \cdot F}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 2e+55)
   (* (- (sqrt (/ 1.0 C))) (sqrt (- F)))
   (* (- (sqrt 2.0)) (sqrt (* (/ -1.0 B) F)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 2e+55) {
		tmp = -sqrt((1.0 / C)) * sqrt(-F);
	} else {
		tmp = -sqrt(2.0) * sqrt(((-1.0 / B) * F));
	}
	return tmp;
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= 2d+55) then
        tmp = -sqrt((1.0d0 / c)) * sqrt(-f)
    else
        tmp = -sqrt(2.0d0) * sqrt((((-1.0d0) / b) * f))
    end if
    code = tmp
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 2e+55) {
		tmp = -Math.sqrt((1.0 / C)) * Math.sqrt(-F);
	} else {
		tmp = -Math.sqrt(2.0) * Math.sqrt(((-1.0 / B) * F));
	}
	return tmp;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	tmp = 0
	if B <= 2e+55:
		tmp = -math.sqrt((1.0 / C)) * math.sqrt(-F)
	else:
		tmp = -math.sqrt(2.0) * math.sqrt(((-1.0 / B) * F))
	return tmp

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 2e+55)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / C))) * sqrt(Float64(-F)));
	else
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(Float64(-1.0 / B) * F)));
	end
	return tmp
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 2e+55)
		tmp = -sqrt((1.0 / C)) * sqrt(-F);
	else
		tmp = -sqrt(2.0) * sqrt(((-1.0 / B) * F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 2e+55], N[((-N[Sqrt[N[(1.0 / C), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[(-F)], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(N[(-1.0 / B), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.00000000000000002e55

   1. Initial program 21.4%

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

   3. Taylor expanded in A around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 21.8%

      \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
   8. Step-by-step derivation

   9. Applied rewrites 21.8%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 21.8%

       \[\leadsto \sqrt{-F} \cdot \left(-\sqrt{\frac{1}{C}}\right) \]

   if 2.00000000000000002e55 < B

   1. Initial program 7.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 25.4%

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

   6. Taylor expanded in B around inf

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

   8. Applied rewrites 50.3%

      \[\leadsto -\sqrt{F \cdot \frac{-1}{B}} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{C}}\right) \cdot \sqrt{-F}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{-1}{B} \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.7% accurate, 12.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.9%

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

3. Taylor expanded in A around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 18.7%

   \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
8. Step-by-step derivation

9. Applied rewrites 18.7%

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

10. Taylor expanded in C around 0

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

12. Applied rewrites 18.7%

    \[\leadsto \sqrt{-F} \cdot \left(-\sqrt{\frac{1}{C}}\right) \]

13. Final simplification 18.7%

    \[\leadsto \left(-\sqrt{\frac{1}{C}}\right) \cdot \sqrt{-F} \]
14. Add Preprocessing

Alternative 9: 34.8% accurate, 13.6× speedup

Math

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

FPCore

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

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt(-F) / -sqrt(C);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.9%

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

3. Taylor expanded in A around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 18.7%

   \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
8. Step-by-step derivation

9. Applied rewrites 18.7%

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

Alternative 10: 27.5% accurate, 14.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.9%

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

3. Taylor expanded in A around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 14.2%

   \[\leadsto -\sqrt{\left(\frac{F}{C} \cdot -0.5\right) \cdot 2} \]
8. Add Preprocessing

Alternative 11: 27.5% accurate, 18.9× speedup

Math

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

FPCore

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

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return -sqrt((-F / C));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.9%

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

3. Taylor expanded in A around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 18.7%

   \[\leadsto -\frac{\sqrt{F \cdot -1}}{\sqrt{C}} \]
8. Step-by-step derivation

9. Applied rewrites 14.2%

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

Reproduce

herbie shell --seed 2024298 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))