Quadratic roots, narrow range

Percentage Accurate: 55.8% → 99.3%

Time: 11.6s

Alternatives: 9

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: 55.8% 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: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 53.1

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

4. Applied rewrites 53.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 53.1

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 53.1

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 53.1

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 53.1

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

6. Applied rewrites 53.1%

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

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}} \]

9. Taylor expanded in a around inf

   \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{a \cdot \left(-4 \cdot c + \frac{{b}^{2}}{a}\right)}}\right) \cdot \left(a \cdot 2\right)} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{a \cdot \left(-4 \cdot c + \frac{{b}^{2}}{a}\right)}}\right) \cdot \left(a \cdot 2\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{a \cdot \left(\color{blue}{c \cdot -4} + \frac{{b}^{2}}{a}\right)}\right) \cdot \left(a \cdot 2\right)} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{a \cdot \color{blue}{\mathsf{fma}\left(c, -4, \frac{{b}^{2}}{a}\right)}}\right) \cdot \left(a \cdot 2\right)} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{a \cdot \mathsf{fma}\left(c, -4, \color{blue}{\frac{{b}^{2}}{a}}\right)}\right) \cdot \left(a \cdot 2\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{a \cdot \mathsf{fma}\left(c, -4, \frac{\color{blue}{b \cdot b}}{a}\right)}\right) \cdot \left(a \cdot 2\right)} \]

    6. lower-*.f64 99.3

       \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{a \cdot \mathsf{fma}\left(c, -4, \frac{\color{blue}{b \cdot b}}{a}\right)}\right) \cdot \left(a \cdot 2\right)} \]

11. Applied rewrites 99.3%

    \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\color{blue}{a \cdot \mathsf{fma}\left(c, -4, \frac{b \cdot b}{a}\right)}}\right) \cdot \left(a \cdot 2\right)} \]

12. Final simplification 99.3%

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

Alternative 2: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot a\right) \cdot \left(--4\right)}{a \cdot \mathsf{fma}\left(b, -4, \frac{c \cdot a}{b} \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.009)
   (/ (- (sqrt (fma b b (* -4.0 (* c a)))) b) (* a 2.0))
   (/ (* (* c a) (- -4.0)) (* a (fma b -4.0 (* (/ (* c a) b) 4.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.009) {
		tmp = (sqrt(fma(b, b, (-4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = ((c * a) * -(-4.0)) / (a * fma(b, -4.0, (((c * a) / b) * 4.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.009)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c * a) * Float64(-(-4.0))) / Float64(a * fma(b, -4.0, Float64(Float64(Float64(c * a) / b) * 4.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.009], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * a), $MachinePrecision] * (--4.0)), $MachinePrecision] / N[(a * N[(b * -4.0 + N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] * 4.0), $MachinePrecision]), $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)) < -0.00899999999999999932

   1. Initial program 78.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 78.5

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

   4. Applied rewrites 78.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 78.5

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 78.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 78.5

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

   6. Applied rewrites 78.5%

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

   if -0.00899999999999999932 < (/.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 45.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 45.1

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

   4. Applied rewrites 45.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 45.1

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 45.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 45.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 45.2

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

   6. Applied rewrites 45.2%

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

   8. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\color{blue}{a \cdot \left(-4 \cdot b + 4 \cdot \frac{a \cdot c}{b}\right)}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\color{blue}{a \cdot \left(-4 \cdot b + 4 \cdot \frac{a \cdot c}{b}\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{a \cdot \left(\color{blue}{b \cdot -4} + 4 \cdot \frac{a \cdot c}{b}\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{a \cdot \color{blue}{\mathsf{fma}\left(b, -4, 4 \cdot \frac{a \cdot c}{b}\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{a \cdot \mathsf{fma}\left(b, -4, \color{blue}{4 \cdot \frac{a \cdot c}{b}}\right)} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{a \cdot \mathsf{fma}\left(b, -4, 4 \cdot \color{blue}{\frac{a \cdot c}{b}}\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{a \cdot \mathsf{fma}\left(b, -4, 4 \cdot \frac{\color{blue}{c \cdot a}}{b}\right)} \]

       7. lower-*.f64 91.3

          \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{a \cdot \mathsf{fma}\left(b, -4, 4 \cdot \frac{\color{blue}{c \cdot a}}{b}\right)} \]

   11. Applied rewrites 91.3%

       \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\color{blue}{a \cdot \mathsf{fma}\left(b, -4, 4 \cdot \frac{c \cdot a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot a\right) \cdot \left(--4\right)}{a \cdot \mathsf{fma}\left(b, -4, \frac{c \cdot a}{b} \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.009)
   (/ (- (sqrt (fma b b (* -4.0 (* c a)))) b) (* a 2.0))
   (- (fma a (/ (* c c) (* b (* b b))) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.009) {
		tmp = (sqrt(fma(b, b, (-4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.009)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.009], N[(N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $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)) < -0.00899999999999999932

   1. Initial program 78.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 78.5

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

   4. Applied rewrites 78.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 78.5

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 78.5

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 78.5

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

   6. Applied rewrites 78.5%

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

   if -0.00899999999999999932 < (/.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 45.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 73.5

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 91.1

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

   8. Applied rewrites 91.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.009)
   (* (/ -0.5 a) (- b (sqrt (fma c (* -4.0 a) (* b b)))))
   (- (fma a (/ (* c c) (* b (* b b))) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.009) {
		tmp = (-0.5 / a) * (b - sqrt(fma(c, (-4.0 * a), (b * b))));
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.009)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(c, Float64(-4.0 * a), Float64(b * b)))));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.009], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(c * N[(-4.0 * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $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)) < -0.00899999999999999932

   1. Initial program 78.5%

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

   4. Applied rewrites 78.4%

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

   if -0.00899999999999999932 < (/.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 45.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 73.5

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 91.1

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

   8. Applied rewrites 91.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.009:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 53.1

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

4. Applied rewrites 53.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 53.1

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 53.1

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 53.1

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 53.1

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

6. Applied rewrites 53.1%

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

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}} \]
9. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{0 - -4 \cdot \left(c \cdot a\right)}}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \]

   2. sub0-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 99.3

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

10. Applied rewrites 99.3%

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

11. Final simplification 99.3%

    \[\leadsto \frac{\left(c \cdot a\right) \cdot 4}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}\right)} \]
12. Add Preprocessing

Alternative 6: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 53.1

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

4. Applied rewrites 53.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 53.1

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 53.1

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 53.1

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 53.1

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

6. Applied rewrites 53.1%

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

8. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)}} \]
9. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right) \cdot \left(a \cdot 2\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{c \cdot \color{blue}{\left(a \cdot -4\right)} + b \cdot b}\right) \cdot \left(a \cdot 2\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4} + b \cdot b}\right) \cdot \left(a \cdot 2\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4 + b \cdot b}\right) \cdot \left(a \cdot 2\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}\right) \cdot \left(a \cdot 2\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \]

   7. lift-*.f64 99.3

      \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\color{blue}{c \cdot a}, -4, b \cdot b\right)}\right) \cdot \left(a \cdot 2\right)} \]

10. Applied rewrites 99.3%

    \[\leadsto \frac{0 - -4 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}\right) \cdot \left(a \cdot 2\right)} \]

11. Final simplification 99.3%

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

Alternative 7: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (- (fma a (/ (* c c) (* b (* b b))) (/ c b))))

C

double code(double a, double b, double c) {
	return -fma(a, ((c * c) / (b * (b * b))), (c / b));
}

Julia

function code(a, b, c)
	return Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)))
end

Wolfram

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

Derivation

1. Initial program 53.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 66.7

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

5. Applied rewrites 66.7%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-out N/A

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

   5. lower-neg.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 84.0

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

8. Applied rewrites 84.0%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)} \]
9. Add Preprocessing

Alternative 8: 81.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ (fma (* c c) (/ a (* b b)) c) (- b)))

C

double code(double a, double b, double c) {
	return fma((c * c), (a / (b * b)), c) / -b;
}

Julia

function code(a, b, c)
	return Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b))
end

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

      \[\leadsto \color{blue}{-1 \cdot \frac{c + \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 83.9

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

5. Applied rewrites 83.9%

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

6. Final simplification 83.9%

   \[\leadsto \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b} \]
7. Add Preprocessing

Alternative 9: 64.1% 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 53.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 66.7

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

5. Applied rewrites 66.7%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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)))