Cubic critical, medium range

Percentage Accurate: 31.8% → 99.3%

Time: 11.7s

Alternatives: 8

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.8% 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

\[\begin{array}{l} \\ \frac{-1}{a \cdot -3} \cdot \frac{c \cdot \left(a \cdot -3\right)}{\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} + b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ -1.0 (* a -3.0))
  (/ (* c (* a -3.0)) (+ (sqrt (fma (* c a) -3.0 (* b b))) b))))

C

double code(double a, double b, double c) {
	return (-1.0 / (a * -3.0)) * ((c * (a * -3.0)) / (sqrt(fma((c * a), -3.0, (b * b))) + b));
}

Julia

function code(a, b, c)
	return Float64(Float64(-1.0 / Float64(a * -3.0)) * Float64(Float64(c * Float64(a * -3.0)) / Float64(sqrt(fma(Float64(c * a), -3.0, Float64(b * b))) + b)))
end

Wolfram

code[a_, b_, c_] := N[(N[(-1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.3%

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

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 29.7%

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

6. Applied rewrites 30.3%

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

   1. lift-neg.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 N/A

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

8. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)}} \cdot \frac{1}{-3 \cdot a} \]
9. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(a \cdot c\right) \cdot -3 + 0}}{\mathsf{neg}\left(\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)\right)} \cdot \frac{1}{-3 \cdot a} \]

   2. +-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 99.3

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

10. Applied rewrites 99.3%

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

11. Final simplification 99.3%

    \[\leadsto \frac{-1}{a \cdot -3} \cdot \frac{c \cdot \left(a \cdot -3\right)}{\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} + b} \]
12. Add Preprocessing

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{a} \cdot \frac{\mathsf{fma}\left(c \cdot a, -3, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ -0.3333333333333333 a)
  (/ (fma (* c a) -3.0 0.0) (- (- b) (sqrt (fma (* c a) -3.0 (* b b)))))))

C

double code(double a, double b, double c) {
	return (-0.3333333333333333 / a) * (fma((c * a), -3.0, 0.0) / (-b - sqrt(fma((c * a), -3.0, (b * b)))));
}

Julia

function code(a, b, c)
	return Float64(Float64(-0.3333333333333333 / a) * Float64(fma(Float64(c * a), -3.0, 0.0) / Float64(Float64(-b) - sqrt(fma(Float64(c * a), -3.0, Float64(b * b))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * -3.0 + 0.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.3%

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

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 29.7%

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

6. Applied rewrites 30.3%

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

   1. lift-neg.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 N/A

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

8. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)}} \cdot \frac{1}{-3 \cdot a} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{\mathsf{neg}\left(\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)\right)} \cdot \color{blue}{\frac{1}{-3 \cdot a}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{\mathsf{neg}\left(\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)\right)} \cdot \frac{1}{\color{blue}{-3 \cdot a}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{\mathsf{neg}\left(\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)\right)} \cdot \color{blue}{\frac{\frac{1}{-3}}{a}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{\mathsf{neg}\left(\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)\right)} \cdot \frac{\color{blue}{\frac{-1}{3}}}{a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{\mathsf{neg}\left(\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)\right)} \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{1}{3}\right)}}{a} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{\mathsf{neg}\left(\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)\right)} \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{a}} \]

   7. metadata-eval 99.0

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)} \cdot \frac{\color{blue}{-0.3333333333333333}}{a} \]

10. Applied rewrites 99.0%

    \[\leadsto \frac{\mathsf{fma}\left(a \cdot c, -3, 0\right)}{-\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b\right)} \cdot \color{blue}{\frac{-0.3333333333333333}{a}} \]

11. Final simplification 99.0%

    \[\leadsto \frac{-0.3333333333333333}{a} \cdot \frac{\mathsf{fma}\left(c \cdot a, -3, 0\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}} \]
12. Add Preprocessing

Alternative 3: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-3}{8} \cdot a}, \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \frac{\color{blue}{c \cdot c}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \color{blue}{c \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \color{blue}{c \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \color{blue}{\frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

   13. lower-*.f64 91.0

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

5. Applied rewrites 91.0%

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

6. Final simplification 91.0%

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

Alternative 4: 90.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}}{b} \]

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{-3}{8} \cdot a}, \frac{{c}^{2}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \frac{\color{blue}{c \cdot c}}{{b}^{2}}, \frac{-1}{2} \cdot c\right)}{b} \]

   8. associate-/l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \color{blue}{c \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, \color{blue}{c \cdot \frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \color{blue}{\frac{c}{{b}^{2}}}, \frac{-1}{2} \cdot c\right)}{b} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot a, c \cdot \frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2} \cdot c\right)}{b} \]

   13. lower-*.f64 91.0

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

5. Applied rewrites 91.0%

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

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 91.0%

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

9. Final simplification 91.0%

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

Alternative 5: 90.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

5. Applied rewrites 93.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

6. Taylor expanded in b around inf

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

8. Applied rewrites 90.7%

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

9. Final simplification 90.7%

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

Alternative 6: 81.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c / b) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 82.2

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

5. Applied rewrites 82.2%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Add Preprocessing

Alternative 7: 80.8% 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 30.3%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot c} \]

5. Applied rewrites 93.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 82.0%

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

Alternative 8: 3.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 30.3%

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

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

   3. +-commutative N/A

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

   4. lift-neg.f64 N/A

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

   5. unsub-neg N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

4. Applied rewrites 29.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. metadata-eval 29.8

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

   14. lift-/.f64 N/A

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

   15. clear-num N/A

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

   16. associate-/r/ N/A

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

   17. lower-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. associate-/r* N/A

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

   21. lower-/.f64 N/A

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

   22. metadata-eval 31.9

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

   23. lift-fma.f64 N/A

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

6. Applied rewrites 31.9%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

   3. mul0-rgt 3.2

      \[\leadsto \color{blue}{0} \]

9. Applied rewrites 3.2%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))