Cubic critical, narrow range

Percentage Accurate: 54.9% → 99.3%

Time: 13.1s

Alternatives: 15

Speedup: 2.9×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(3 \cdot a\right) \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{\left(-3\right) \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ (* (* 3.0 a) c) (+ (sqrt (fma (* c -3.0) a (* b b))) b)) (* (- 3.0) a)))

C

double code(double a, double b, double c) {
	return (((3.0 * a) * c) / (sqrt(fma((c * -3.0), a, (b * b))) + b)) / (-3.0 * a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(3.0 * a) * c) / Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b)) / Float64(Float64(-3.0) * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[((-3.0) * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 55.3

       \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{0.3333333333333333} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-neg.f64 N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 55.3

       \[\leadsto \frac{1}{a} \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)}\right) \]

4. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\right)} \]

6. Applied rewrites 56.5%

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

7. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(3 \cdot a\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. associate-/r* N/A

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

    6. lower-/.f64 N/A

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

    \[\leadsto \frac{\frac{\left(3 \cdot a\right) \cdot c}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}{\left(-3\right) \cdot a} \]
13. Add Preprocessing

Alternative 2: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(\frac{c}{b} \cdot a\right)\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.01)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/
    0.3333333333333333
    (/ (fma -0.6666666666666666 b (* 0.5 (* (/ c b) a))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.01) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = 0.3333333333333333 / (fma(-0.6666666666666666, b, (0.5 * ((c / b) * a))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.01)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(0.3333333333333333 / Float64(fma(-0.6666666666666666, b, Float64(0.5 * Float64(Float64(c / b) * a))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[(-0.6666666666666666 * b + N[(0.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0100000000000000002

   1. Initial program 81.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 81.2

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

   4. Applied rewrites 81.2%

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

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 45.8%

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

   4. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 88.8

         \[\leadsto \frac{0.3333333333333333}{\frac{\mathsf{fma}\left(-0.6666666666666666, b, \left(a \cdot \color{blue}{\frac{c}{b}}\right) \cdot 0.5\right)}{c}} \]

   7. Applied rewrites 88.8%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{\mathsf{fma}\left(-0.6666666666666666, b, \left(a \cdot \frac{c}{b}\right) \cdot 0.5\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{\mathsf{fma}\left(-0.6666666666666666, b, 0.5 \cdot \left(\frac{c}{b} \cdot a\right)\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.01)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.01) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.01)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0100000000000000002

   1. Initial program 81.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 81.2

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

   4. Applied rewrites 81.2%

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

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 45.8%

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

   4. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.7

         \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \color{blue}{\frac{a}{b}} \cdot 0.5\right)} \]

   7. Applied rewrites 88.7%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.01)
   (* (/ (- (sqrt (fma b b (* (* a c) -3.0))) b) a) 0.3333333333333333)
   (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.01) {
		tmp = ((sqrt(fma(b, b, ((a * c) * -3.0))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.01)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -3.0))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0100000000000000002

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 81.0

         \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 81.0

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{c \cdot -3}, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333 \]

   6. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.1

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)} - b}{a} \cdot 0.3333333333333333 \]

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 81.1

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right)} \cdot -3\right)} - b}{a} \cdot 0.3333333333333333 \]

   8. Applied rewrites 81.1%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}} - b}{a} \cdot 0.3333333333333333 \]

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 45.8%

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

   4. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.7

         \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \color{blue}{\frac{a}{b}} \cdot 0.5\right)} \]

   7. Applied rewrites 88.7%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.01:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.01:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.01)
   (* (- (sqrt (fma (* c -3.0) a (* b b))) b) (/ 0.3333333333333333 a))
   (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.01) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.01)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0100000000000000002

   1. Initial program 81.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 81.0

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 81.0

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)} \]

   4. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)} \]

   if -0.0100000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 45.8%

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

   4. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.7

         \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \color{blue}{\frac{a}{b}} \cdot 0.5\right)} \]

   7. Applied rewrites 88.7%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.01:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.3333333333333333}{a} \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (/ 0.3333333333333333 a) (* (* 3.0 a) c))
  (- (- b) (sqrt (fma (* c -3.0) a (* b b))))))

C

double code(double a, double b, double c) {
	return ((0.3333333333333333 / a) * ((3.0 * a) * c)) / (-b - sqrt(fma((c * -3.0), a, (b * b))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(0.3333333333333333 / a) * Float64(Float64(3.0 * a) * c)) / Float64(Float64(-b) - sqrt(fma(Float64(c * -3.0), a, Float64(b * b)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 55.3

       \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{0.3333333333333333} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-neg.f64 N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 55.3

       \[\leadsto \frac{1}{a} \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)}\right) \]

4. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\right)} \]

6. Applied rewrites 56.5%

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

7. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(3 \cdot a\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f64 N/A

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

11. Applied rewrites 99.1%

    \[\leadsto \color{blue}{\frac{\left(\left(a \cdot 3\right) \cdot c\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}}} \]

12. Final simplification 99.1%

    \[\leadsto \frac{\frac{0.3333333333333333}{a} \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}} \]
13. Add Preprocessing

Alternative 7: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{-\left(a \cdot c\right) \cdot 3}{\left(\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot 3\right) \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (* (* a c) 3.0)) (* (* (+ (sqrt (fma (* c -3.0) a (* b b))) b) 3.0) a)))

C

double code(double a, double b, double c) {
	return -((a * c) * 3.0) / (((sqrt(fma((c * -3.0), a, (b * b))) + b) * 3.0) * a);
}

Julia

function code(a, b, c)
	return Float64(Float64(-Float64(Float64(a * c) * 3.0)) / Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b) * 3.0) * a))
end

Wolfram

code[a_, b_, c_] := N[((-N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]) / N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * 3.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 55.3

       \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{0.3333333333333333} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-neg.f64 N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 55.3

       \[\leadsto \frac{1}{a} \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)}\right) \]

4. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\right)} \]

6. Applied rewrites 56.5%

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

7. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(3 \cdot a\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 99.1

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lift-*.f64 99.1

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

11. Applied rewrites 99.1%

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

12. Final simplification 99.1%

    \[\leadsto \frac{-\left(a \cdot c\right) \cdot 3}{\left(\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot 3\right) \cdot a} \]
13. Add Preprocessing

Alternative 8: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{-\left(a \cdot c\right) \cdot 3}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot \left(3 \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (* (* a c) 3.0)) (* (+ (sqrt (fma (* c -3.0) a (* b b))) b) (* 3.0 a))))

C

double code(double a, double b, double c) {
	return -((a * c) * 3.0) / ((sqrt(fma((c * -3.0), a, (b * b))) + b) * (3.0 * a));
}

Julia

function code(a, b, c)
	return Float64(Float64(-Float64(Float64(a * c) * 3.0)) / Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b) * Float64(3.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[((-N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision]) / N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 55.3

       \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{0.3333333333333333} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-neg.f64 N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 55.3

       \[\leadsto \frac{1}{a} \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)}\right) \]

4. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\right)} \]

6. Applied rewrites 56.5%

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

7. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(3 \cdot a\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

10. Final simplification 99.1%

    \[\leadsto \frac{-\left(a \cdot c\right) \cdot 3}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot \left(3 \cdot a\right)} \]
11. Add Preprocessing

Alternative 9: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\left(a \cdot c\right) \cdot 3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot \left(3 \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (* a c) 3.0) (* (- (- b) (sqrt (fma b b (* (* a c) -3.0)))) (* 3.0 a))))

C

double code(double a, double b, double c) {
	return ((a * c) * 3.0) / ((-b - sqrt(fma(b, b, ((a * c) * -3.0)))) * (3.0 * a));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * c) * 3.0) / Float64(Float64(Float64(-b) - sqrt(fma(b, b, Float64(Float64(a * c) * -3.0)))) * Float64(3.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * c), $MachinePrecision] * 3.0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 55.3

       \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{0.3333333333333333} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-neg.f64 N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 55.3

       \[\leadsto \frac{1}{a} \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)}\right) \]

4. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\right)} \]

6. Applied rewrites 56.5%

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

7. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(3 \cdot a\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

    1. lift-fma.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lower-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 99.1

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

    11. lift-*.f64 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 99.1

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

11. Applied rewrites 99.1%

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

12. Final simplification 99.1%

    \[\leadsto \frac{\left(a \cdot c\right) \cdot 3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot \left(3 \cdot a\right)} \]
13. Add Preprocessing

Alternative 10: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(-0.3333333333333333\right) \cdot \frac{\left(3 \cdot a\right) \cdot c}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (- 0.3333333333333333)
  (/ (* (* 3.0 a) c) (* (+ (sqrt (fma (* c -3.0) a (* b b))) b) a))))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 * (((3.0 * a) * c) / ((sqrt(fma((c * -3.0), a, (b * b))) + b) * a));
}

Julia

function code(a, b, c)
	return Float64(Float64(-0.3333333333333333) * Float64(Float64(Float64(3.0 * a) * c) / Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b) * a)))
end

Wolfram

code[a_, b_, c_] := N[((-0.3333333333333333) * N[(N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision] / N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 55.3

       \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{0.3333333333333333} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-neg.f64 N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 55.3

       \[\leadsto \frac{1}{a} \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)}\right) \]

4. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\right)} \]

6. Applied rewrites 56.5%

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

7. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(3 \cdot a\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

    1. lift-/.f64 N/A

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

    2. div-inv N/A

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

    3. metadata-eval N/A

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

    4. associate-*r/ N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

11. Applied rewrites 99.0%

    \[\leadsto \color{blue}{\frac{\left(a \cdot 3\right) \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}\right) \cdot a} \cdot 0.3333333333333333} \]

12. Final simplification 99.0%

    \[\leadsto \left(-0.3333333333333333\right) \cdot \frac{\left(3 \cdot a\right) \cdot c}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot a} \]
13. Add Preprocessing

Alternative 11: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(-\left(3 \cdot a\right) \cdot c\right) \cdot \frac{0.3333333333333333}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (- (* (* 3.0 a) c))
  (/ 0.3333333333333333 (* (+ (sqrt (fma (* c -3.0) a (* b b))) b) a))))

C

double code(double a, double b, double c) {
	return -((3.0 * a) * c) * (0.3333333333333333 / ((sqrt(fma((c * -3.0), a, (b * b))) + b) * a));
}

Julia

function code(a, b, c)
	return Float64(Float64(-Float64(Float64(3.0 * a) * c)) * Float64(0.3333333333333333 / Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) + b) * a)))
end

Wolfram

code[a_, b_, c_] := N[((-N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]) * N[(0.3333333333333333 / N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 55.3

       \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{0.3333333333333333} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \]

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-neg.f64 N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 55.3

       \[\leadsto \frac{1}{a} \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right)}\right) \]

4. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\right)} \]

6. Applied rewrites 56.5%

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

7. Taylor expanded in c around 0

   \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(3 \cdot a\right) \cdot \left(\left(\mathsf{neg}\left(b\right)\right) - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

    1. lift-/.f64 N/A

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

    2. div-inv N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

11. Applied rewrites 99.0%

    \[\leadsto \color{blue}{\frac{0.3333333333333333}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)}\right) \cdot a} \cdot \left(\left(a \cdot 3\right) \cdot c\right)} \]

12. Final simplification 99.0%

    \[\leadsto \left(-\left(3 \cdot a\right) \cdot c\right) \cdot \frac{0.3333333333333333}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right) \cdot a} \]
13. Add Preprocessing

Alternative 12: 82.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5))))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

4. Applied rewrites 55.3%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

5. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 81.4

      \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \color{blue}{\frac{a}{b}} \cdot 0.5\right)} \]

7. Applied rewrites 81.4%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}} \]
8. Add Preprocessing

Alternative 13: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{c}{b \cdot b} \cdot a, -0.375, -0.5\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (fma (* (/ c (* b b)) a) -0.375 -0.5) c) b))

C

double code(double a, double b, double c) {
	return (fma(((c / (b * b)) * a), -0.375, -0.5) * c) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(Float64(c / Float64(b * b)) * a), -0.375, -0.5) * c) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -0.375 + -0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

5. Applied rewrites 86.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625 \cdot \left(a \cdot a\right), \left(c \cdot c\right) \cdot \frac{c}{{b}^{4}}, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, -0.5 \cdot c\right)\right)}{b}} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 80.8%

   \[\leadsto \frac{\mathsf{fma}\left(a \cdot \frac{c}{b \cdot b}, -0.375, -0.5\right) \cdot c}{b} \]

9. Final simplification 80.8%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{c}{b \cdot b} \cdot a, -0.375, -0.5\right) \cdot c}{b} \]
10. Add Preprocessing

Alternative 14: 64.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (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 = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.4

      \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

5. Applied rewrites 64.4%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

6. Final simplification 64.4%

   \[\leadsto -0.5 \cdot \frac{c}{b} \]
7. Add Preprocessing

Alternative 15: 64.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{b} \cdot c \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ -0.5 b) c))

C

double code(double a, double b, double c) {
	return (-0.5 / 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 = ((-0.5d0) / b) * c
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 / b) * c;
}

Python

def code(a, b, c):
	return (-0.5 / b) * c

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 / b) * c)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 / b) * c;
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 / b), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.4

      \[\leadsto \color{blue}{\frac{c}{b}} \cdot -0.5 \]

5. Applied rewrites 64.4%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 64.3%

   \[\leadsto c \cdot \color{blue}{\frac{-0.5}{b}} \]

8. Final simplification 64.3%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))