ABCF->ab-angle b

Percentage Accurate: 19.1% → 49.5%

Time: 18.0s

Alternatives: 7

Speedup: 13.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 49.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{-0.5}{C} \cdot \left(\sqrt{F \cdot -4} \cdot \sqrt{C}\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{C} \cdot \sqrt{C \cdot \left(F \cdot -4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_0) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B 2.0)))))
   (if (<= t_1 (- INFINITY))
     (* (/ -0.5 C) (* (sqrt (* F -4.0)) (sqrt C)))
     (if (<= t_1 -1e-193)
       (/
        (sqrt
         (*
          (* (fma B B (* C (* A -4.0))) (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))))
        (fma B (- B) (* A (* 4.0 C))))
       (* (/ -0.5 C) (sqrt (* C (* F -4.0))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B, 2.0) - t_0) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B, 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-0.5 / C) * (sqrt((F * -4.0)) * sqrt(C));
	} else if (t_1 <= -1e-193) {
		tmp = sqrt(((fma(B, B, (C * (A * -4.0))) * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B)))))) / fma(B, -B, (A * (4.0 * C)));
	} else {
		tmp = (-0.5 / C) * sqrt((C * (F * -4.0)));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B ^ 2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-0.5 / C) * Float64(sqrt(Float64(F * -4.0)) * sqrt(C)));
	elseif (t_1 <= -1e-193)
		tmp = Float64(sqrt(Float64(Float64(fma(B, B, Float64(C * Float64(A * -4.0))) * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))) / fma(B, Float64(-B), Float64(A * Float64(4.0 * C))));
	else
		tmp = Float64(Float64(-0.5 / C) * sqrt(Float64(C * Float64(F * -4.0))));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 / C), $MachinePrecision] * N[(N[Sqrt[N[(F * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-193], N[(N[Sqrt[N[(N[(N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / C), $MachinePrecision] * N[Sqrt[N[(C * N[(F * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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 3.4%

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 39.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 39.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 39.7%

      \[\leadsto \frac{-0.5}{C} \cdot \color{blue}{\sqrt{F \cdot \left(C \cdot -4\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 66.7%

       \[\leadsto \frac{-0.5}{C} \cdot \left(\sqrt{F \cdot -4} \cdot \color{blue}{\sqrt{C}}\right) \]

   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))) < -1e-193

   1. Initial program 97.9%

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

   4. Applied rewrites 97.9%

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

   if -1e-193 < (/.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 3.6%

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 35.7

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 35.7%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 36.2%

      \[\leadsto \frac{-0.5}{C} \cdot \color{blue}{\sqrt{F \cdot \left(C \cdot -4\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 36.2%

       \[\leadsto \frac{-0.5}{C} \cdot \sqrt{\left(F \cdot -4\right) \cdot C} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{-0.5}{C} \cdot \left(\sqrt{F \cdot -4} \cdot \sqrt{C}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-193}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{C} \cdot \sqrt{C \cdot \left(F \cdot -4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B}^{2}}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+162}:\\ \;\;\;\;\frac{-0.5}{C} \cdot \left(\sqrt{F \cdot -4} \cdot \sqrt{C}\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{C} \cdot \sqrt{C \cdot \left(F \cdot -4\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_0) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B 2.0)))))
   (if (<= t_1 -4e+162)
     (* (/ -0.5 C) (* (sqrt (* F -4.0)) (sqrt C)))
     (if (<= t_1 -1e-170)
       (*
        (sqrt
         (/
          (* F (- (+ A C) (sqrt (fma B B (* (- A C) (- A C))))))
          (fma (* A C) -4.0 (* B B))))
        (- (sqrt 2.0)))
       (* (/ -0.5 C) (sqrt (* C (* F -4.0))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B, 2.0) - t_0) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B, 2.0));
	double tmp;
	if (t_1 <= -4e+162) {
		tmp = (-0.5 / C) * (sqrt((F * -4.0)) * sqrt(C));
	} else if (t_1 <= -1e-170) {
		tmp = sqrt(((F * ((A + C) - sqrt(fma(B, B, ((A - C) * (A - C)))))) / fma((A * C), -4.0, (B * B)))) * -sqrt(2.0);
	} else {
		tmp = (-0.5 / C) * sqrt((C * (F * -4.0)));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -4e+162)
		tmp = Float64(Float64(-0.5 / C) * Float64(sqrt(Float64(F * -4.0)) * sqrt(C)));
	elseif (t_1 <= -1e-170)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B, B, Float64(Float64(A - C) * Float64(A - C)))))) / fma(Float64(A * C), -4.0, Float64(B * B)))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-0.5 / C) * sqrt(Float64(C * Float64(F * -4.0))));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+162], N[(N[(-0.5 / C), $MachinePrecision] * N[(N[Sqrt[N[(F * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-170], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B * B + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(-0.5 / C), $MachinePrecision] * N[Sqrt[N[(C * N[(F * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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))) < -3.9999999999999998e162

   1. Initial program 9.6%

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

   4. Applied rewrites 2.2%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 46.3

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 46.3%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.0%

      \[\leadsto \frac{-0.5}{C} \cdot \color{blue}{\sqrt{F \cdot \left(C \cdot -4\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.7%

       \[\leadsto \frac{-0.5}{C} \cdot \left(\sqrt{F \cdot -4} \cdot \color{blue}{\sqrt{C}}\right) \]

   if -3.9999999999999998e162 < (/.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))) < -9.99999999999999983e-171

   1. Initial program 97.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 93.8%

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

   if -9.99999999999999983e-171 < (/.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 7.9%

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

   4. Applied rewrites 2.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 34.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 34.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 34.6%

      \[\leadsto \frac{-0.5}{C} \cdot \color{blue}{\sqrt{F \cdot \left(C \cdot -4\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 34.6%

       \[\leadsto \frac{-0.5}{C} \cdot \sqrt{\left(F \cdot -4\right) \cdot C} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{+162}:\\ \;\;\;\;\frac{-0.5}{C} \cdot \left(\sqrt{F \cdot -4} \cdot \sqrt{C}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{C} \cdot \sqrt{C \cdot \left(F \cdot -4\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(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))))