Cubic critical, narrow range

Percentage Accurate: 55.1% → 99.6%

Time: 17.8s

Alternatives: 9

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: 55.1% 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

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

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(Float64(c * 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[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 58.3%

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

3. Simplified 58.3%

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

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

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

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

6. Applied egg-rr 59.9%

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

7. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 99.6%

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

9. Simplified 99.6%

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

    1. sub0-neg N/A

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

    2. neg-lowering-neg.f64 99.6%

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

11. Applied egg-rr 99.6%

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

    1. associate-*r* N/A

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

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

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

    3. *-lowering-*.f64 99.6%

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

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

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

Alternative 2: 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 58.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 58.3%

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

3. Simplified 58.3%

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

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

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

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

6. Applied egg-rr 59.9%

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

7. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 99.6%

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

9. Simplified 99.6%

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

    1. sub0-neg N/A

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

    2. neg-lowering-neg.f64 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 3: 88.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c / ((a * (((-1.125 * (a * (0.0 - (c * c)))) / (b * (b * b))) - (-1.5 * (c / 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 = c / ((a * ((((-1.125d0) * (a * (0.0d0 - (c * c)))) / (b * (b * b))) - ((-1.5d0) * (c / b)))) - (b * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 58.3%

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

3. Simplified 58.3%

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

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

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

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

6. Applied egg-rr 59.9%

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

7. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 99.6%

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

9. Simplified 99.6%

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

    1. sub0-neg N/A

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

    2. neg-lowering-neg.f64 99.6%

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in a around 0

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

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

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

    2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

    8. associate-*r/ N/A

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

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

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

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

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

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

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

    12. unpow2 N/A

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

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

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

    14. cube-mult N/A

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

    15. unpow2 N/A

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

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

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

    17. unpow2 N/A

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

    18. *-lowering-*.f64 85.8%

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

14. Simplified 85.8%

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

15. Final simplification 85.8%

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

Alternative 4: 88.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c / (((a * (((-1.125 * (a * (0.0 - (c * c)))) / (b * (b * b))) - (-1.5 * (c / 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 / (((a * ((((-1.125d0) * (a * (0.0d0 - (c * c)))) / (b * (b * b))) - ((-1.5d0) * (c / b)))) - b) - b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 58.3%

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

3. Simplified 58.3%

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

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

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

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

6. Applied egg-rr 59.9%

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

7. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 99.6%

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

9. Simplified 99.6%

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

    1. sub0-neg N/A

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

    2. neg-lowering-neg.f64 99.6%

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

    6. associate-*r/ N/A

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

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

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

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

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

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

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

    10. unpow2 N/A

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

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

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

    12. cube-mult N/A

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

    13. unpow2 N/A

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

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

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

    15. unpow2 N/A

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

    16. *-lowering-*.f64 85.7%

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

14. Simplified 85.7%

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

15. Final simplification 85.7%

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

Alternative 5: 82.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-1.5 \cdot \frac{0 - c \cdot a}{b} - b \cdot 2} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 58.3%

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

3. Simplified 58.3%

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

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

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

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

6. Applied egg-rr 59.9%

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

7. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 99.6%

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

9. Simplified 99.6%

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

    1. sub0-neg N/A

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

    2. neg-lowering-neg.f64 99.6%

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

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

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

    3. *-commutative N/A

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

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

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

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

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

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

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

    7. *-lowering-*.f64 78.9%

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

14. Simplified 78.9%

    \[\leadsto \frac{-c}{\color{blue}{b \cdot 2 + -1.5 \cdot \frac{a \cdot c}{b}}} \]

15. Final simplification 78.9%

    \[\leadsto \frac{c}{-1.5 \cdot \frac{0 - c \cdot a}{b} - b \cdot 2} \]
16. Add Preprocessing

Alternative 6: 82.3% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 58.3%

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

3. Simplified 58.3%

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

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

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

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

6. Applied egg-rr 59.9%

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

7. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 99.6%

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

9. Simplified 99.6%

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

    1. sub0-neg N/A

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

    2. neg-lowering-neg.f64 99.6%

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

11. Applied egg-rr 99.6%

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

12. Taylor expanded in c around 0

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

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

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

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

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

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

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

    4. *-lowering-*.f64 78.9%

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

14. Simplified 78.9%

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

15. Final simplification 78.9%

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

Alternative 7: 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 58.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 58.3%

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

3. Simplified 58.3%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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 61.5%

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

7. Simplified 61.5%

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

Alternative 8: 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 58.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 58.3%

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

3. Simplified 58.3%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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 61.5%

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

7. Simplified 61.5%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
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 61.4%

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

9. Applied egg-rr 61.4%

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

10. Final simplification 61.4%

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

Alternative 9: 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 58.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 58.3%

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

3. Simplified 58.3%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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. div-inv N/A

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

   3. div-inv N/A

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

   4. prod-diff N/A

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

   5. associate-/r/ N/A

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

   6. clear-num N/A

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

   7. fmm-def N/A

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

   8. div-inv N/A

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

   9. div-sub N/A

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

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

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

6. Applied egg-rr 57.0%

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

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.2%

      \[\leadsto 0 \]

9. Simplified 3.2%

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

Reproduce

herbie shell --seed 2024152 
(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)))