Quadratic roots, narrow range

Percentage Accurate: 55.3% → 92.0%

Time: 14.9s

Alternatives: 16

Speedup: 3.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

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

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

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

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0, \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + t\_1, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, -2 \cdot \mathsf{fma}\left(c, -\frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}, -0.5 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}\right), \frac{a}{b}\right), -b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -17.0)
     (/ (fma t_1 t_0 (* (- b) (* b b))) (* (* a 2.0) (fma b (+ b t_1) t_0)))
     (/
      1.0
      (/
       (fma
        c
        (fma
         c
         (*
          -2.0
          (fma
           c
           (- (/ (* a (* a a)) (pow b 5.0)))
           (* -0.5 (/ (* a a) (* b (* b b))))))
         (/ a b))
        (- b))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -17.0) {
		tmp = fma(t_1, t_0, (-b * (b * b))) / ((a * 2.0) * fma(b, (b + t_1), t_0));
	} else {
		tmp = 1.0 / (fma(c, fma(c, (-2.0 * fma(c, -((a * (a * a)) / pow(b, 5.0)), (-0.5 * ((a * a) / (b * (b * b)))))), (a / b)), -b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -17.0)
		tmp = Float64(fma(t_1, t_0, Float64(Float64(-b) * Float64(b * b))) / Float64(Float64(a * 2.0) * fma(b, Float64(b + t_1), t_0)));
	else
		tmp = Float64(1.0 / Float64(fma(c, fma(c, Float64(-2.0 * fma(c, Float64(-Float64(Float64(a * Float64(a * a)) / (b ^ 5.0))), Float64(-0.5 * Float64(Float64(a * a) / Float64(b * Float64(b * b)))))), Float64(a / b)), Float64(-b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -17.0], N[(N[(t$95$1 * t$95$0 + N[((-b) * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b * N[(b + t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * N[(c * N[(-2.0 * N[(c * (-N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]) + N[(-0.5 * N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -17

   1. Initial program 85.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      5. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]

      6. div-sub N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

   4. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}} \]

   6. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), -b \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}} \]

   if -17 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 51.8

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 51.8

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 51.8

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 51.8

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   7. Taylor expanded in c around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + c \cdot \left(c \cdot \left(-2 \cdot \left(c \cdot \left(-1 \cdot \frac{a \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{1}{2} \cdot \frac{{a}^{2}}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-1}{8} \cdot \frac{b \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a} + \frac{{a}^{3}}{{b}^{5}}\right)\right)\right) + -2 \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{1}{2} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + \frac{a}{b}\right)}{c}}} \]

   9. Applied rewrites 93.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, -2 \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.125, \frac{b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)}{a}, \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - a \cdot \frac{\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5}{b \cdot b}, \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right), \frac{a}{b}\right), -b\right)}{c}}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, -2 \cdot \mathsf{fma}\left(c, \frac{\left(\frac{-5}{2} \cdot {a}^{3} + {a}^{3}\right) - \frac{-1}{2} \cdot {a}^{3}}{{b}^{5}}, \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot \frac{-1}{2}\right), \frac{a}{b}\right), \mathsf{neg}\left(b\right)\right)}{c}} \]
   11. Step-by-step derivation

   12. Applied rewrites 93.2%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, -2 \cdot \mathsf{fma}\left(c, \frac{-a \cdot \left(a \cdot a\right)}{{b}^{5}}, \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right), \frac{a}{b}\right), -b\right)}{c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c, -2 \cdot \mathsf{fma}\left(c, -\frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}, -0.5 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}\right), \frac{a}{b}\right), -b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0, \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + t\_1, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-5, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)}{a \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot t\_2\right)\right)}, \frac{\left(c \cdot -2\right) \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{b \cdot t\_2}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))) (t_1 (sqrt t_0)) (t_2 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -17.0)
     (/ (fma t_1 t_0 (* (- b) (* b b))) (* (* a 2.0) (fma b (+ b t_1) t_0)))
     (-
      (/
       (-
        (fma
         -5.0
         (/
          (* (* a (* a (* a a))) (* (* c c) (* c c)))
          (* a (* b (* (* b b) t_2))))
         (/ (* (* c -2.0) (* (* a a) (* c c))) (* b t_2)))
        (/ (* a (* c c)) (* b b)))
       b)
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -17.0) {
		tmp = fma(t_1, t_0, (-b * (b * b))) / ((a * 2.0) * fma(b, (b + t_1), t_0));
	} else {
		tmp = ((fma(-5.0, (((a * (a * (a * a))) * ((c * c) * (c * c))) / (a * (b * ((b * b) * t_2)))), (((c * -2.0) * ((a * a) * (c * c))) / (b * t_2))) - ((a * (c * c)) / (b * b))) / b) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -17.0)
		tmp = Float64(fma(t_1, t_0, Float64(Float64(-b) * Float64(b * b))) / Float64(Float64(a * 2.0) * fma(b, Float64(b + t_1), t_0)));
	else
		tmp = Float64(Float64(Float64(fma(-5.0, Float64(Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(Float64(c * c) * Float64(c * c))) / Float64(a * Float64(b * Float64(Float64(b * b) * t_2)))), Float64(Float64(Float64(c * -2.0) * Float64(Float64(a * a) * Float64(c * c))) / Float64(b * t_2))) - Float64(Float64(a * Float64(c * c)) / Float64(b * b))) / b) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -17.0], N[(N[(t$95$1 * t$95$0 + N[((-b) * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b * N[(b + t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-5.0 * N[(N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b * N[(N[(b * b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * -2.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -17

   1. Initial program 85.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      5. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]

      6. div-sub N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

   4. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}} \]

   6. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), -b \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}} \]

   if -17 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.25 \cdot \left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{20}{a \cdot {b}^{6}}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)\right)}{b}} \]

   7. Applied rewrites 92.9%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot 0.05} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right) - c}{b} \]

   9. Applied rewrites 93.0%

      \[\leadsto \frac{\mathsf{fma}\left(-5, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)}{a \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, \frac{\left(c \cdot -2\right) \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b} - \color{blue}{\frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-5, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)}{a \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, \frac{\left(c \cdot -2\right) \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right) - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}}{b} - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot \left(c \cdot c\right)\right)\\ t_1 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ t_2 := \sqrt{t\_1}\\ t_3 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_1, \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + t\_2, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{c \cdot t\_0}{b \cdot t\_3}, \frac{-0.25 \cdot \left(t\_0 \cdot t\_0\right)}{\left(a \cdot \left(t\_3 \cdot t\_3\right)\right) \cdot 0.05}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* a (* c c))))
        (t_1 (fma a (* c -4.0) (* b b)))
        (t_2 (sqrt t_1))
        (t_3 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -17.0)
     (/ (fma t_2 t_1 (* (- b) (* b b))) (* (* a 2.0) (fma b (+ b t_2) t_1)))
     (/
      (-
       (fma
        -2.0
        (/ (* c t_0) (* b t_3))
        (/ (* -0.25 (* t_0 t_0)) (* (* a (* t_3 t_3)) 0.05)))
       (fma (* c c) (/ a (* b b)) c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = a * (a * (c * c));
	double t_1 = fma(a, (c * -4.0), (b * b));
	double t_2 = sqrt(t_1);
	double t_3 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -17.0) {
		tmp = fma(t_2, t_1, (-b * (b * b))) / ((a * 2.0) * fma(b, (b + t_2), t_1));
	} else {
		tmp = (fma(-2.0, ((c * t_0) / (b * t_3)), ((-0.25 * (t_0 * t_0)) / ((a * (t_3 * t_3)) * 0.05))) - fma((c * c), (a / (b * b)), c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(a * Float64(c * c)))
	t_1 = fma(a, Float64(c * -4.0), Float64(b * b))
	t_2 = sqrt(t_1)
	t_3 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -17.0)
		tmp = Float64(fma(t_2, t_1, Float64(Float64(-b) * Float64(b * b))) / Float64(Float64(a * 2.0) * fma(b, Float64(b + t_2), t_1)));
	else
		tmp = Float64(Float64(fma(-2.0, Float64(Float64(c * t_0) / Float64(b * t_3)), Float64(Float64(-0.25 * Float64(t_0 * t_0)) / Float64(Float64(a * Float64(t_3 * t_3)) * 0.05))) - fma(Float64(c * c), Float64(a / Float64(b * b)), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -17.0], N[(N[(t$95$2 * t$95$1 + N[((-b) * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b * N[(b + t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * N[(N[(c * t$95$0), $MachinePrecision] / N[(b * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(a * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.05), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -17

   1. Initial program 85.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      5. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]

      6. div-sub N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

   4. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}} \]

   6. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), -b \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}} \]

   if -17 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.25 \cdot \left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{20}{a \cdot {b}^{6}}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)\right)}{b}} \]

   7. Applied rewrites 92.9%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot 0.05}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot 0.05}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0, \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + t\_1, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-5, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)}{a \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot t\_2\right)\right)}, \frac{\left(c \cdot -2\right) \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{b \cdot t\_2}\right) - \mathsf{fma}\left(a \cdot c, \frac{c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))) (t_1 (sqrt t_0)) (t_2 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -17.0)
     (/ (fma t_1 t_0 (* (- b) (* b b))) (* (* a 2.0) (fma b (+ b t_1) t_0)))
     (/
      (-
       (fma
        -5.0
        (/
         (* (* a (* a (* a a))) (* (* c c) (* c c)))
         (* a (* b (* (* b b) t_2))))
        (/ (* (* c -2.0) (* (* a a) (* c c))) (* b t_2)))
       (fma (* a c) (/ c (* b b)) c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -17.0) {
		tmp = fma(t_1, t_0, (-b * (b * b))) / ((a * 2.0) * fma(b, (b + t_1), t_0));
	} else {
		tmp = (fma(-5.0, (((a * (a * (a * a))) * ((c * c) * (c * c))) / (a * (b * ((b * b) * t_2)))), (((c * -2.0) * ((a * a) * (c * c))) / (b * t_2))) - fma((a * c), (c / (b * b)), c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -17.0)
		tmp = Float64(fma(t_1, t_0, Float64(Float64(-b) * Float64(b * b))) / Float64(Float64(a * 2.0) * fma(b, Float64(b + t_1), t_0)));
	else
		tmp = Float64(Float64(fma(-5.0, Float64(Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(Float64(c * c) * Float64(c * c))) / Float64(a * Float64(b * Float64(Float64(b * b) * t_2)))), Float64(Float64(Float64(c * -2.0) * Float64(Float64(a * a) * Float64(c * c))) / Float64(b * t_2))) - fma(Float64(a * c), Float64(c / Float64(b * b)), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -17.0], N[(N[(t$95$1 * t$95$0 + N[((-b) * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b * N[(b + t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-5.0 * N[(N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(b * N[(N[(b * b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * -2.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -17

   1. Initial program 85.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      5. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]

      6. div-sub N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

   4. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}} \]

   6. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), -b \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}} \]

   if -17 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.25 \cdot \left(\left({a}^{4} \cdot {c}^{4}\right) \cdot \frac{20}{a \cdot {b}^{6}}\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)\right)}{b}} \]

   7. Applied rewrites 92.9%

      \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{-0.25 \cdot \left(\left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot 0.05} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right) - c}{b} \]

   9. Applied rewrites 92.9%

      \[\leadsto \frac{\mathsf{fma}\left(-5, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)}{a \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, \frac{\left(c \cdot -2\right) \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right) - \mathsf{fma}\left(a \cdot c, \frac{c}{b \cdot b}, c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-5, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)}{a \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, \frac{\left(c \cdot -2\right) \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right) - \mathsf{fma}\left(a \cdot c, \frac{c}{b \cdot b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_0, \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + t\_1, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot -2, -0.5 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}, \frac{a}{b}\right), -b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -4.0) (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -17.0)
     (/ (fma t_1 t_0 (* (- b) (* b b))) (* (* a 2.0) (fma b (+ b t_1) t_0)))
     (/
      1.0
      (/
       (fma
        c
        (fma (* c -2.0) (* -0.5 (/ (* a a) (* b (* b b)))) (/ a b))
        (- b))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (c * -4.0), (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -17.0) {
		tmp = fma(t_1, t_0, (-b * (b * b))) / ((a * 2.0) * fma(b, (b + t_1), t_0));
	} else {
		tmp = 1.0 / (fma(c, fma((c * -2.0), (-0.5 * ((a * a) / (b * (b * b)))), (a / b)), -b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -17.0)
		tmp = Float64(fma(t_1, t_0, Float64(Float64(-b) * Float64(b * b))) / Float64(Float64(a * 2.0) * fma(b, Float64(b + t_1), t_0)));
	else
		tmp = Float64(1.0 / Float64(fma(c, fma(Float64(c * -2.0), Float64(-0.5 * Float64(Float64(a * a) / Float64(b * Float64(b * b)))), Float64(a / b)), Float64(-b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -17.0], N[(N[(t$95$1 * t$95$0 + N[((-b) * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b * N[(b + t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * N[(N[(c * -2.0), $MachinePrecision] * N[(-0.5 * N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -17

   1. Initial program 85.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      5. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]

      6. div-sub N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}} \]

   4. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}} \]

   6. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), -b \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}} \]

   if -17 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 51.8

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 51.8

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 51.8

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 51.8

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   7. Taylor expanded in c around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + c \cdot \left(-2 \cdot \left(c \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{1}{2} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + \frac{a}{b}\right)}{c}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + c \cdot \left(-2 \cdot \left(c \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{1}{2} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + \frac{a}{b}\right)}{c}}} \]

   9. Applied rewrites 90.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-2 \cdot c, \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{a}{b}\right), -b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right), \left(-b\right) \cdot \left(b \cdot b\right)\right)}{\left(a \cdot 2\right) \cdot \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot -2, -0.5 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}, \frac{a}{b}\right), -b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(t\_0 - b \cdot b\right)}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot -2, -0.5 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}, \frac{a}{b}\right), -b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -17.0)
     (/ (* (/ 0.5 a) (- t_0 (* b b))) (+ b (sqrt t_0)))
     (/
      1.0
      (/
       (fma
        c
        (fma (* c -2.0) (* -0.5 (/ (* a a) (* b (* b b)))) (/ a b))
        (- b))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -17.0) {
		tmp = ((0.5 / a) * (t_0 - (b * b))) / (b + sqrt(t_0));
	} else {
		tmp = 1.0 / (fma(c, fma((c * -2.0), (-0.5 * ((a * a) / (b * (b * b)))), (a / b)), -b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -17.0)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(t_0 - Float64(b * b))) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(1.0 / Float64(fma(c, fma(Float64(c * -2.0), Float64(-0.5 * Float64(Float64(a * a) / Float64(b * Float64(b * b)))), Float64(a / b)), Float64(-b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -17.0], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * N[(N[(c * -2.0), $MachinePrecision] * N[(-0.5 * N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -17

   1. Initial program 85.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 85.3

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 85.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 85.4

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 85.4

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 85.4

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   8. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

   if -17 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 51.8

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 51.8

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 51.8

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 51.8

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   7. Taylor expanded in c around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + c \cdot \left(-2 \cdot \left(c \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{1}{2} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + \frac{a}{b}\right)}{c}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + c \cdot \left(-2 \cdot \left(c \cdot \left(-1 \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{1}{2} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right) + \frac{a}{b}\right)}{c}}} \]

   9. Applied rewrites 90.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-2 \cdot c, \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{a}{b}\right), -b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot -2, -0.5 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}, \frac{a}{b}\right), -b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(t\_0 - b \cdot b\right)}{b + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, -0.5 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{1}{b}\right), \frac{b}{-c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -17.0)
     (/ (* (/ 0.5 a) (- t_0 (* b b))) (+ b (sqrt t_0)))
     (/
      1.0
      (fma
       a
       (fma (* a -2.0) (* -0.5 (/ c (* b (* b b)))) (/ 1.0 b))
       (/ b (- c)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -17.0) {
		tmp = ((0.5 / a) * (t_0 - (b * b))) / (b + sqrt(t_0));
	} else {
		tmp = 1.0 / fma(a, fma((a * -2.0), (-0.5 * (c / (b * (b * b)))), (1.0 / b)), (b / -c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -17.0)
		tmp = Float64(Float64(Float64(0.5 / a) * Float64(t_0 - Float64(b * b))) / Float64(b + sqrt(t_0)));
	else
		tmp = Float64(1.0 / fma(a, fma(Float64(a * -2.0), Float64(-0.5 * Float64(c / Float64(b * Float64(b * b)))), Float64(1.0 / b)), Float64(b / Float64(-c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -17.0], N[(N[(N[(0.5 / a), $MachinePrecision] * N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(N[(a * -2.0), $MachinePrecision] * N[(-0.5 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[(b / (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -17

   1. Initial program 85.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 85.3

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 85.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 85.4

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 85.4

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 85.4

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   8. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

   if -17 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 51.8

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 51.8

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 51.8

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 51.8

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right) + -1 \cdot \frac{b}{c}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, -2 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}, -1 \cdot \frac{b}{c}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\left(-2 \cdot a\right) \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)} + \frac{1}{b}, -1 \cdot \frac{b}{c}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(-2 \cdot a, -1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}, \frac{1}{b}\right)}, -1 \cdot \frac{b}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{-2 \cdot a}, -1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      6. distribute-rgt-out N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \color{blue}{\frac{c}{{b}^{3}} \cdot \left(-1 + \frac{1}{2}\right)}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{{b}^{3}} \cdot \color{blue}{\frac{-1}{2}}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \color{blue}{\frac{c}{{b}^{3}} \cdot \frac{-1}{2}}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \color{blue}{\frac{c}{{b}^{3}}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      10. cube-mult N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{\color{blue}{b \cdot \left(b \cdot b\right)}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \color{blue}{{b}^{2}}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{\color{blue}{b \cdot {b}^{2}}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \color{blue}{\left(b \cdot b\right)}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \color{blue}{\left(b \cdot b\right)}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-1}{2}, \color{blue}{\frac{1}{b}}\right), -1 \cdot \frac{b}{c}\right)} \]

   9. Applied rewrites 90.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{1}{b}\right), \frac{b}{-c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, -0.5 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{1}{b}\right), \frac{b}{-c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, -0.5 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{1}{b}\right), \frac{b}{-c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -17.0)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (/
      1.0
      (fma
       a
       (fma (* a -2.0) (* -0.5 (/ c (* b (* b b)))) (/ 1.0 b))
       (/ b (- c)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -17.0) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = 1.0 / fma(a, fma((a * -2.0), (-0.5 * (c / (b * (b * b)))), (1.0 / b)), (b / -c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -17.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(1.0 / fma(a, fma(Float64(a * -2.0), Float64(-0.5 * Float64(c / Float64(b * Float64(b * b)))), Float64(1.0 / b)), Float64(b / Float64(-c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -17.0], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(a * N[(N[(a * -2.0), $MachinePrecision] * N[(-0.5 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[(b / (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -17

   1. Initial program 85.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 85.3

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 85.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 85.4

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 85.4

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 85.4

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   8. Applied rewrites 87.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}} \]

   if -17 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 51.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 51.8

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 51.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 51.8

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 51.8

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 51.8

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right) + -1 \cdot \frac{b}{c}}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, -2 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}, -1 \cdot \frac{b}{c}\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\left(-2 \cdot a\right) \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}\right)} + \frac{1}{b}, -1 \cdot \frac{b}{c}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(-2 \cdot a, -1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}, \frac{1}{b}\right)}, -1 \cdot \frac{b}{c}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{-2 \cdot a}, -1 \cdot \frac{c}{{b}^{3}} + \frac{1}{2} \cdot \frac{c}{{b}^{3}}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      6. distribute-rgt-out N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \color{blue}{\frac{c}{{b}^{3}} \cdot \left(-1 + \frac{1}{2}\right)}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{{b}^{3}} \cdot \color{blue}{\frac{-1}{2}}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \color{blue}{\frac{c}{{b}^{3}} \cdot \frac{-1}{2}}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \color{blue}{\frac{c}{{b}^{3}}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      10. cube-mult N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{\color{blue}{b \cdot \left(b \cdot b\right)}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \color{blue}{{b}^{2}}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{\color{blue}{b \cdot {b}^{2}}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \color{blue}{\left(b \cdot b\right)}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \color{blue}{\left(b \cdot b\right)}} \cdot \frac{-1}{2}, \frac{1}{b}\right), -1 \cdot \frac{b}{c}\right)} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot \frac{-1}{2}, \color{blue}{\frac{1}{b}}\right), -1 \cdot \frac{b}{c}\right)} \]

   9. Applied rewrites 90.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{1}{b}\right), \frac{b}{-c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -17:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, -0.5 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{1}{b}\right), \frac{b}{-c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(a \cdot a, -2 \cdot \frac{c}{b \cdot b}, -a\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b))))
   (if (<= b 210.0)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (*
      c
      (fma
       c
       (/ (fma (* a a) (* -2.0 (/ c (* b b))) (- a)) (* b (* b b)))
       (/ -1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (b <= 210.0) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = c * fma(c, (fma((a * a), (-2.0 * (c / (b * b))), -a) / (b * (b * b))), (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 210.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(c * fma(c, Float64(fma(Float64(a * a), Float64(-2.0 * Float64(c / Float64(b * b))), Float64(-a)) / Float64(b * Float64(b * b))), Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 210.0], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(c * N[(N[(N[(a * a), $MachinePrecision] * N[(-2.0 * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 210

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 79.3

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 79.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 79.2

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 79.2

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 79.2

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   8. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}} \]

   if 210 < b

   1. Initial program 42.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{-2 \cdot \left({a}^{2} \cdot c\right)}{{b}^{5}}} + -1 \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto c \cdot \left(c \cdot \left(\frac{\color{blue}{\left(-2 \cdot {a}^{2}\right) \cdot c}}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\frac{-2 \cdot {a}^{2}}{{b}^{5}} \cdot c} + -1 \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto c \cdot \left(c \cdot \left(\color{blue}{\left(-2 \cdot \frac{{a}^{2}}{{b}^{5}}\right)} \cdot c + -1 \cdot \frac{a}{{b}^{3}}\right) + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(-1 \cdot \frac{a}{{b}^{3}} + \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c\right)} + \left(\mathsf{neg}\left(\frac{1}{b}\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c\right) + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{b}}\right) \]

      9. metadata-eval N/A

         \[\leadsto c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c\right) + \frac{\color{blue}{-1}}{b}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot \frac{a}{{b}^{3}} + \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}}\right) \cdot c, \frac{-1}{b}\right)} \]

   5. Applied rewrites 95.0%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, \frac{-2 \cdot \left(a \cdot \left(a \cdot c\right)\right)}{{b}^{5}} - \frac{a}{b \cdot \left(b \cdot b\right)}, \frac{-1}{b}\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} - a}{\color{blue}{{b}^{3}}}, \frac{-1}{b}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 95.0%

      \[\leadsto c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(a \cdot a, \frac{c}{b \cdot b} \cdot -2, -a\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(a \cdot a, -2 \cdot \frac{c}{b \cdot b}, -a\right)}{b \cdot \left(b \cdot b\right)}, \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b))))
   (if (<= b 210.0)
     (/ (- t_0 (* b b)) (* (* a 2.0) (+ b (sqrt t_0))))
     (/ 1.0 (/ (- (/ (* a c) b) b) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (b <= 210.0) {
		tmp = (t_0 - (b * b)) / ((a * 2.0) * (b + sqrt(t_0)));
	} else {
		tmp = 1.0 / ((((a * c) / b) - b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (b <= 210.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(a * 2.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(a * c) / b) - b) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 210.0], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 210

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 79.3

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 79.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 79.2

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 79.2

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 79.2

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   8. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}} \]

   if 210 < b

   1. Initial program 42.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 42.9

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 42.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 42.9

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 42.9

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 42.9

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   7. Taylor expanded in c around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} + -1 \cdot b}}{c}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{\frac{a \cdot c}{b} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{c}} \]

      4. unsub-neg N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b}} - b}{c}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

      8. lower-*.f64 91.7

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

   9. Applied rewrites 91.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{c \cdot a}{b} - b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 210:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 255:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 255.0)
   (/ (- (sqrt (fma b b (* a (* c -4.0)))) b) (* a 2.0))
   (/ 1.0 (/ (- (/ (* a c) b) b) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 255.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((((a * c) / b) - b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 255.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(a * c) / b) - b) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 255.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 255

   1. Initial program 79.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right)}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(4 \cdot a\right)\right) \cdot c}\right)}}{2 \cdot a} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{4 \cdot a}\right)\right) \cdot c\right)}}{2 \cdot a} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{a \cdot 4}\right)\right) \cdot c\right)}}{2 \cdot a} \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(4\right)\right)\right)} \cdot c\right)}}{2 \cdot a} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot c\right)}\right)}}{2 \cdot a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot c\right)}\right)}}{2 \cdot a} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot c\right)}\right)}}{2 \cdot a} \]

      13. metadata-eval 79.5

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(\color{blue}{-4} \cdot c\right)\right)}}{2 \cdot a} \]

   4. Applied rewrites 79.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, a \cdot \left(-4 \cdot c\right)\right)}}}{2 \cdot a} \]

   if 255 < b

   1. Initial program 42.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 42.7

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 42.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 42.7

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 42.7

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 42.7

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   7. Taylor expanded in c around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} + -1 \cdot b}}{c}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{\frac{a \cdot c}{b} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{c}} \]

      4. unsub-neg N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b}} - b}{c}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

      8. lower-*.f64 91.8

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

   9. Applied rewrites 91.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{c \cdot a}{b} - b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 255:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 255:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 255.0)
   (/ (- (sqrt (fma c (* a -4.0) (* b b))) b) (* a 2.0))
   (/ 1.0 (/ (- (/ (* a c) b) b) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 255.0) {
		tmp = (sqrt(fma(c, (a * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((((a * c) / b) - b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 255.0)
		tmp = Float64(Float64(sqrt(fma(c, Float64(a * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(a * c) / b) - b) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 255.0], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 255

   1. Initial program 79.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{2 \cdot a} \]

      4. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]

      5. lower--.f64 79.2

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}{2 \cdot a} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}{2 \cdot a} \]

      7. sub-neg N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}} - b}{2 \cdot a} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}} - b}{2 \cdot a} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b} - b}{2 \cdot a} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{c \cdot \left(4 \cdot a\right)}\right)\right) + b \cdot b} - b}{2 \cdot a} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(\mathsf{neg}\left(4 \cdot a\right)\right)} + b \cdot b} - b}{2 \cdot a} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(4 \cdot a\right), b \cdot b\right)}} - b}{2 \cdot a} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(c, \mathsf{neg}\left(\color{blue}{4 \cdot a}\right), b \cdot b\right)} - b}{2 \cdot a} \]

      14. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(c, \mathsf{neg}\left(\color{blue}{a \cdot 4}\right), b \cdot b\right)} - b}{2 \cdot a} \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(\mathsf{neg}\left(4\right)\right)}, b \cdot b\right)} - b}{2 \cdot a} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(\mathsf{neg}\left(4\right)\right)}, b \cdot b\right)} - b}{2 \cdot a} \]

      17. metadata-eval 79.2

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-4}, b \cdot b\right)} - b}{2 \cdot a} \]

   4. Applied rewrites 79.2%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}{2 \cdot a} \]

   if 255 < b

   1. Initial program 42.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 42.7

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 42.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 42.7

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 42.7

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 42.7

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   7. Taylor expanded in c around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} + -1 \cdot b}}{c}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{\frac{a \cdot c}{b} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{c}} \]

      4. unsub-neg N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b}} - b}{c}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

      8. lower-*.f64 91.8

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

   9. Applied rewrites 91.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{c \cdot a}{b} - b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 255:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 255:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 255.0)
   (* (/ -0.5 a) (- b (sqrt (fma c (* a -4.0) (* b b)))))
   (/ 1.0 (/ (- (/ (* a c) b) b) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 255.0) {
		tmp = (-0.5 / a) * (b - sqrt(fma(c, (a * -4.0), (b * b))));
	} else {
		tmp = 1.0 / ((((a * c) / b) - b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 255.0)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(c, Float64(a * -4.0), Float64(b * b)))));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(a * c) / b) - b) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 255.0], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 255

   1. Initial program 79.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 79.2%

      \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \]

   if 255 < b

   1. Initial program 42.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

      11. metadata-eval 42.7

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

   4. Applied rewrites 42.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

      2. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      6. lower-/.f64 42.7

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      9. lower-*.f64 42.7

         \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      14. *-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      15. associate-*r* N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

      19. unsub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

      20. lower--.f64 42.7

          \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   6. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   7. Taylor expanded in c around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} + -1 \cdot b}}{c}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{\frac{a \cdot c}{b} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{c}} \]

      4. unsub-neg N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b}} - b}{c}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

      8. lower-*.f64 91.8

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

   9. Applied rewrites 91.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{c \cdot a}{b} - b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 255:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (/ (- (/ (* a c) b) b) c)))

C

double code(double a, double b, double c) {
	return 1.0 / ((((a * c) / b) - b) / c);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / ((((a * c) / b) - b) / c)
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((((a * c) / b) - b) / c);
}

Python

def code(a, b, c):
	return 1.0 / ((((a * c) / b) - b) / c)

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(Float64(a * c) / b) - b) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((((a * c) / b) - b) / c);
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   2. sub-neg N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

   11. metadata-eval 56.2

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

4. Applied rewrites 56.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   6. lower-/.f64 56.2

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   9. lower-*.f64 56.2

      \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

   11. +-commutative N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

   12. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   14. *-commutative N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   15. associate-*r* N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   17. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   18. lift-neg.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

   19. unsub-neg N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   20. lower--.f64 56.2

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

6. Applied rewrites 56.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

7. Taylor expanded in c around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot b + \frac{a \cdot c}{b}}{c}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} + -1 \cdot b}}{c}} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{1}{\frac{\frac{a \cdot c}{b} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{c}} \]

   4. unsub-neg N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b} - b}}{c}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{a \cdot c}{b}} - b}{c}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

   8. lower-*.f64 81.1

      \[\leadsto \frac{1}{\frac{\frac{\color{blue}{c \cdot a}}{b} - b}{c}} \]

9. Applied rewrites 81.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{c \cdot a}{b} - b}{c}}} \]

10. Final simplification 81.1%

    \[\leadsto \frac{1}{\frac{\frac{a \cdot c}{b} - b}{c}} \]
11. Add Preprocessing

Alternative 15: 82.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (- (/ a b) (/ b c))))

C

double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / ((a / b) - (b / c))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((a / b) - (b / c));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

   2. sub-neg N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right) + b \cdot b}}}{2 \cdot a} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right) + b \cdot b}}{2 \cdot a} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right) + b \cdot b}}{2 \cdot a} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b}}{2 \cdot a} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(a \cdot c\right) \cdot 4}\right)\right) + b \cdot b}}{2 \cdot a} \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(\mathsf{neg}\left(4\right)\right)} + b \cdot b}}{2 \cdot a} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, \mathsf{neg}\left(4\right), b \cdot b\right)}}}{2 \cdot a} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(\color{blue}{a \cdot c}, \mathsf{neg}\left(4\right), b \cdot b\right)}}{2 \cdot a} \]

   11. metadata-eval 56.2

       \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, \color{blue}{-4}, b \cdot b\right)}}{2 \cdot a} \]

4. Applied rewrites 56.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\frac{2 \cdot a}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   6. lower-/.f64 56.2

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{2 \cdot a}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   9. lower-*.f64 56.2

      \[\leadsto \frac{1}{\frac{\color{blue}{a \cdot 2}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}} \]

   11. +-commutative N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + \left(\mathsf{neg}\left(b\right)\right)}}} \]

   12. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   14. *-commutative N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -4 + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   15. associate-*r* N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   17. lift-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} + \left(\mathsf{neg}\left(b\right)\right)}} \]

   18. lift-neg.f64 N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}} \]

   19. unsub-neg N/A

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

   20. lower--.f64 56.2

       \[\leadsto \frac{1}{\frac{a \cdot 2}{\color{blue}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

6. Applied rewrites 56.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}}} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + \frac{a}{b}}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} + -1 \cdot \frac{b}{c}}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{1}{\frac{a}{b} + \color{blue}{\left(\mathsf{neg}\left(\frac{b}{c}\right)\right)}} \]

   3. unsub-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{b}} - \frac{b}{c}} \]

   6. lower-/.f64 81.1

      \[\leadsto \frac{1}{\frac{a}{b} - \color{blue}{\frac{b}{c}}} \]

9. Applied rewrites 81.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
10. Add Preprocessing

Alternative 16: 64.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c (- b)))

C

double code(double a, double b, double c) {
	return c / -b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / -b
end function

Java

public static double code(double a, double b, double c) {
	return c / -b;
}

Python

def code(a, b, c):
	return c / -b

Julia

function code(a, b, c)
	return Float64(c / Float64(-b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / -b;
end

Wolfram

code[a_, b_, c_] := N[(c / (-b)), $MachinePrecision]

Derivation

1. Initial program 56.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   4. lower-neg.f64 63.5

      \[\leadsto \frac{c}{\color{blue}{-b}} \]

5. Applied rewrites 63.5%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))