Cubic critical, narrow range

Percentage Accurate: 55.4% → 99.6%

Time: 18.1s

Alternatives: 8

Speedup: 23.2×

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: 55.4% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / ((0.0d0 - b) - sqrt(((b * b) + (c * (a * (-3.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return c / ((0.0 - b) - Math.sqrt(((b * b) + (c * (a * -3.0)))));
}

Python

def code(a, b, c):
	return c / ((0.0 - b) - math.sqrt(((b * b) + (c * (a * -3.0)))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c / ((0.0 - b) - sqrt(((b * b) + (c * (a * -3.0)))));
end

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 52.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 52.5%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. associate-/r* N/A

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

   4. frac-times N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b \cdot b\right) \cdot \frac{1}{3}\right), \color{blue}{\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} + b\right) \cdot a\right)}\right) \]

6. Applied egg-rr 53.8%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right), a\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(a \cdot c\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)}, a\right)\right) \]

   2. *-lowering-*.f64 99.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(a, c\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)}\right), a\right)\right) \]

9. Simplified 99.3%

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. times-frac N/A

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

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1 \cdot a}{a}\right), \color{blue}{\left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)}\right) \]

    5. neg-mul-1 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(a\right)}{a}\right), \left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(a\right)\right), a\right), \left(\frac{\color{blue}{c}}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    7. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(0 - a\right), a\right), \left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    8. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \color{blue}{\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right) \]

    10. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right)\right) \]

    11. rem-square-sqrt N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right)\right)\right) \]

    12. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right)\right)\right) \]

    13. rem-square-sqrt N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right) \]

    14. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

    17. *-lowering-*.f64 99.6%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 99.6%

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

    1. sub0-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(a\right)}{a}\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

    2. distribute-frac-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{a}{a}\right)\right), \mathsf{/.f64}\left(\color{blue}{c}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

    3. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

    4. metadata-eval 99.6%

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\color{blue}{c}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

    \[\leadsto \frac{c}{\left(0 - b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}} \]
15. Add Preprocessing

Alternative 2: 88.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{a}{a} \cdot \frac{c}{a \cdot \left(\frac{c \cdot -1.5}{0 - b} - \frac{-1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right) - b \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ a a)
  (/
   c
   (-
    (*
     a
     (- (/ (* c -1.5) (- 0.0 b)) (/ (* -1.125 (* a (* c c))) (* b (* b b)))))
    (* b 2.0)))))

C

double code(double a, double b, double c) {
	return (a / a) * (c / ((a * (((c * -1.5) / (0.0 - b)) - ((-1.125 * (a * (c * c))) / (b * (b * b))))) - (b * 2.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a / a) * (c / ((a * (((c * (-1.5d0)) / (0.0d0 - b)) - (((-1.125d0) * (a * (c * c))) / (b * (b * b))))) - (b * 2.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (a / a) * (c / ((a * (((c * -1.5) / (0.0 - b)) - ((-1.125 * (a * (c * c))) / (b * (b * b))))) - (b * 2.0)));
}

Python

def code(a, b, c):
	return (a / a) * (c / ((a * (((c * -1.5) / (0.0 - b)) - ((-1.125 * (a * (c * c))) / (b * (b * b))))) - (b * 2.0)))

Julia

function code(a, b, c)
	return Float64(Float64(a / a) * Float64(c / Float64(Float64(a * Float64(Float64(Float64(c * -1.5) / Float64(0.0 - b)) - Float64(Float64(-1.125 * Float64(a * Float64(c * c))) / Float64(b * Float64(b * b))))) - Float64(b * 2.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a / a) * (c / ((a * (((c * -1.5) / (0.0 - b)) - ((-1.125 * (a * (c * c))) / (b * (b * b))))) - (b * 2.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(a / a), $MachinePrecision] * N[(c / N[(N[(a * N[(N[(N[(c * -1.5), $MachinePrecision] / N[(0.0 - b), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.125 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 52.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 52.5%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. associate-/r* N/A

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

   4. frac-times N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b \cdot b\right) \cdot \frac{1}{3}\right), \color{blue}{\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} + b\right) \cdot a\right)}\right) \]

6. Applied egg-rr 53.8%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right), a\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(a \cdot c\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)}, a\right)\right) \]

   2. *-lowering-*.f64 99.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(a, c\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)}\right), a\right)\right) \]

9. Simplified 99.3%

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. times-frac N/A

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

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1 \cdot a}{a}\right), \color{blue}{\left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)}\right) \]

    5. neg-mul-1 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(a\right)}{a}\right), \left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(a\right)\right), a\right), \left(\frac{\color{blue}{c}}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    7. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(0 - a\right), a\right), \left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    8. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \color{blue}{\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right) \]

    10. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right)\right) \]

    11. rem-square-sqrt N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right)\right)\right) \]

    12. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right)\right)\right) \]

    13. rem-square-sqrt N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right) \]

    14. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

    17. *-lowering-*.f64 99.6%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 99.6%

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

12. Taylor expanded in a around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \color{blue}{\left(2 \cdot b + a \cdot \left(\frac{-3}{2} \cdot \frac{c}{b} + \frac{-9}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)}\right)\right) \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \left(a \cdot \left(\frac{-3}{2} \cdot \frac{c}{b} + \frac{-9}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + \color{blue}{2 \cdot b}\right)\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\left(a \cdot \left(\frac{-3}{2} \cdot \frac{c}{b} + \frac{-9}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\right) \]

14. Simplified 90.6%

    \[\leadsto \frac{0 - a}{a} \cdot \frac{c}{\color{blue}{a \cdot \left(\frac{c \cdot -1.5}{b} + \frac{-1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right) + b \cdot 2}} \]

15. Final simplification 90.6%

    \[\leadsto \frac{a}{a} \cdot \frac{c}{a \cdot \left(\frac{c \cdot -1.5}{0 - b} - \frac{-1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right) - b \cdot 2} \]
16. Add Preprocessing

Alternative 3: 82.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \left(0 - \frac{a}{a}\right) \cdot \frac{c}{b \cdot 2 + \frac{-1.5 \cdot \left(c \cdot a\right)}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (- 0.0 (/ a a)) (/ c (+ (* b 2.0) (/ (* -1.5 (* c a)) b)))))

C

double code(double a, double b, double c) {
	return (0.0 - (a / a)) * (c / ((b * 2.0) + ((-1.5 * (c * a)) / 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 = (0.0d0 - (a / a)) * (c / ((b * 2.0d0) + (((-1.5d0) * (c * a)) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return (0.0 - (a / a)) * (c / ((b * 2.0) + ((-1.5 * (c * a)) / b)));
}

Python

def code(a, b, c):
	return (0.0 - (a / a)) * (c / ((b * 2.0) + ((-1.5 * (c * a)) / b)))

Julia

function code(a, b, c)
	return Float64(Float64(0.0 - Float64(a / a)) * Float64(c / Float64(Float64(b * 2.0) + Float64(Float64(-1.5 * Float64(c * a)) / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (0.0 - (a / a)) * (c / ((b * 2.0) + ((-1.5 * (c * a)) / b)));
end

Wolfram

code[a_, b_, c_] := N[(N[(0.0 - N[(a / a), $MachinePrecision]), $MachinePrecision] * N[(c / N[(N[(b * 2.0), $MachinePrecision] + N[(N[(-1.5 * N[(c * a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 52.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 52.5%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. associate-/r* N/A

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

   4. frac-times N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b \cdot b\right) \cdot \frac{1}{3}\right), \color{blue}{\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} + b\right) \cdot a\right)}\right) \]

6. Applied egg-rr 53.8%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right), a\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \left(a \cdot c\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)}, a\right)\right) \]

   2. *-lowering-*.f64 99.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{*.f64}\left(a, c\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)}\right), a\right)\right) \]

9. Simplified 99.3%

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. times-frac N/A

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

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1 \cdot a}{a}\right), \color{blue}{\left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)}\right) \]

    5. neg-mul-1 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(a\right)}{a}\right), \left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(a\right)\right), a\right), \left(\frac{\color{blue}{c}}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    7. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(0 - a\right), a\right), \left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    8. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \left(\frac{c}{b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \color{blue}{\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right) \]

    10. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right)\right) \]

    11. rem-square-sqrt N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \left(\sqrt{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)\right)\right)\right) \]

    12. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right)\right)\right) \]

    13. rem-square-sqrt N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right) \]

    14. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

    17. *-lowering-*.f64 99.6%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right)\right)\right)\right) \]

11. Applied egg-rr 99.6%

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

12. Taylor expanded in c around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \color{blue}{\left(\frac{-3}{2} \cdot \frac{a \cdot c}{b} + 2 \cdot b\right)}\right)\right) \]
13. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{-3}{2} \cdot \frac{a \cdot c}{b}\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{-3}{2} \cdot \left(a \cdot c\right)}{b}\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-3}{2} \cdot \left(a \cdot c\right)\right), b\right), \left(\color{blue}{2} \cdot b\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(a \cdot c\right) \cdot \frac{-3}{2}\right), b\right), \left(2 \cdot b\right)\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), \frac{-3}{2}\right), b\right), \left(2 \cdot b\right)\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c \cdot a\right), \frac{-3}{2}\right), b\right), \left(2 \cdot b\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{2}\right), b\right), \left(2 \cdot b\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{2}\right), b\right), \left(b \cdot \color{blue}{2}\right)\right)\right)\right) \]

    9. *-lowering-*.f64 85.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, a\right), a\right), \mathsf{/.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-3}{2}\right), b\right), \mathsf{*.f64}\left(b, \color{blue}{2}\right)\right)\right)\right) \]

14. Simplified 85.1%

    \[\leadsto \frac{0 - a}{a} \cdot \frac{c}{\color{blue}{\frac{\left(c \cdot a\right) \cdot -1.5}{b} + b \cdot 2}} \]

15. Final simplification 85.1%

    \[\leadsto \left(0 - \frac{a}{a}\right) \cdot \frac{c}{b \cdot 2 + \frac{-1.5 \cdot \left(c \cdot a\right)}{b}} \]
16. Add Preprocessing

Alternative 4: 81.9% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (-0.5d0)) + (((a * (c * c)) * (-0.375d0)) / (b * b))) / b
end function

Java

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

Python

def code(a, b, c):
	return ((c * -0.5) + (((a * (c * c)) * -0.375) / (b * b))) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) + (((a * (c * c)) * -0.375) / (b * b))) / b;
end

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 52.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 52.5%

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

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

7. Simplified 92.9%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{\left(-0.5625 \cdot c\right) \cdot \left(c \cdot c\right)}{{b}^{5}} + \frac{\left(-0.16666666666666666 \cdot a\right) \cdot \frac{{c}^{4} \cdot 6.328125}{{b}^{6}}}{b}\right)\right)} \]

8. Taylor expanded in b around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right), \color{blue}{b}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot c\right), \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right), b\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \left(\frac{\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)}{{b}^{2}}\right)\right), b\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot \left(a \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \left(a \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(a, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(a, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right)\right), b\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right)\right), b\right) \]

   11. *-lowering-*.f64 84.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right)\right), b\right) \]

10. Simplified 84.4%

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

11. Final simplification 84.4%

    \[\leadsto \frac{c \cdot -0.5 + \frac{\left(a \cdot \left(c \cdot c\right)\right) \cdot -0.375}{b \cdot b}}{b} \]
12. Add Preprocessing

Alternative 5: 81.8% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-0.5d0) / b) + (((c * a) * (-0.375d0)) / (b * (b * b))))
end function

Java

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

Python

def code(a, b, c):
	return c * ((-0.5 / b) + (((c * a) * -0.375) / (b * (b * b))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.5 / b) + (((c * a) * -0.375) / (b * (b * b))));
end

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 52.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 52.5%

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

5. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)}\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right) + \color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right), \color{blue}{\left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)}\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{b}\right)\right), \left(\left(\color{blue}{\frac{-3}{8}} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{b}\right)\right), \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   11. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{b}\right), \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \left(\left(\color{blue}{\frac{-3}{8}} \cdot \frac{a}{{b}^{3}}\right) \cdot c\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c\right)\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c\right)\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{\color{blue}{{b}^{3}}}\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{\color{blue}{b}}^{3}}\right)\right)\right) \]

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), \mathsf{/.f64}\left(\left(\frac{-3}{8} \cdot \left(a \cdot c\right)\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

7. Simplified 84.3%

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

8. Final simplification 84.3%

   \[\leadsto c \cdot \left(\frac{-0.5}{b} + \frac{\left(c \cdot a\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right) \]
9. Add Preprocessing

Alternative 6: 64.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 52.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 52.5%

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

5. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

   4. *-lowering-*.f64 66.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

7. Simplified 66.8%

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

Alternative 7: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 52.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 52.5%

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

5. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

   4. *-lowering-*.f64 66.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

7. Simplified 66.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \color{blue}{c}\right) \]

   4. /-lowering-/.f64 66.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), c\right) \]

9. Applied egg-rr 66.8%

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

10. Final simplification 66.8%

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

Alternative 8: 3.2% accurate, 116.0× 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 52.5%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 52.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 52.5%

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

   1. div-sub N/A

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

   2. frac-sub N/A

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

   3. swap-sqr N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \left(3 \cdot a\right) - \left(3 \cdot a\right) \cdot b}{9 \cdot \left(\color{blue}{a} \cdot a\right)} \]

   5. metadata-eval N/A

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

   6. swap-sqr N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. div-inv N/A

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

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \left(3 \cdot a\right) - \left(3 \cdot a\right) \cdot b\right), \color{blue}{\left(\frac{1}{\left(a \cdot -3\right) \cdot \left(a \cdot -3\right)}\right)}\right) \]

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \left(a \cdot 3\right) + \left(a \cdot -3\right) \cdot b\right) \cdot \frac{1}{9 \cdot \left(a \cdot a\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{9} \cdot \frac{-3 \cdot b + 3 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \left(-3 \cdot b + 3 \cdot b\right)}{\color{blue}{a}} \]

   2. distribute-rgt-out N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \left(b \cdot \left(-3 + 3\right)\right)}{a} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \left(b \cdot 0\right)}{a} \]

   4. mul0-rgt N/A

      \[\leadsto \frac{\frac{1}{9} \cdot 0}{a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{0}{a} \]

   6. /-lowering-/.f64 3.2%

      \[\leadsto \mathsf{/.f64}\left(0, \color{blue}{a}\right) \]

9. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
10. Step-by-step derivation

    1. div0 3.2%

       \[\leadsto 0 \]

11. Applied egg-rr 3.2%

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

Reproduce

herbie shell --seed 2024161 
(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)))