Cubic critical, wide range

Percentage Accurate: 17.3% → 97.6%

Time: 22.4s

Alternatives: 12

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.3% 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: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \frac{\left(c \cdot c\right) \cdot \left(-1.0546875 \cdot \left(a \cdot a\right)\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot -0.5625 + \left(b \cdot b\right) \cdot -0.375\right)}{{b}^{7}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (/ (* c -0.5) b)
  (*
   a
   (*
    (* c c)
    (/
     (+
      (* (* c c) (* -1.0546875 (* a a)))
      (* (* b b) (+ (* (* c a) -0.5625) (* (* b b) -0.375))))
     (pow b 7.0))))))

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((c * c) * ((((c * c) * (-1.0546875 * (a * a))) + ((b * b) * (((c * a) * -0.5625) + ((b * b) * -0.375)))) / pow(b, 7.0))));
}

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 * (-0.5d0)) / b) + (a * ((c * c) * ((((c * c) * ((-1.0546875d0) * (a * a))) + ((b * b) * (((c * a) * (-0.5625d0)) + ((b * b) * (-0.375d0))))) / (b ** 7.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((c * c) * ((((c * c) * (-1.0546875 * (a * a))) + ((b * b) * (((c * a) * -0.5625) + ((b * b) * -0.375)))) / Math.pow(b, 7.0))));
}

Python

def code(a, b, c):
	return ((c * -0.5) / b) + (a * ((c * c) * ((((c * c) * (-1.0546875 * (a * a))) + ((b * b) * (((c * a) * -0.5625) + ((b * b) * -0.375)))) / math.pow(b, 7.0))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(c * c) * Float64(Float64(Float64(Float64(c * c) * Float64(-1.0546875 * Float64(a * a))) + Float64(Float64(b * b) * Float64(Float64(Float64(c * a) * -0.5625) + Float64(Float64(b * b) * -0.375)))) / (b ^ 7.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + (a * ((c * c) * ((((c * c) * (-1.0546875 * (a * a))) + ((b * b) * (((c * a) * -0.5625) + ((b * b) * -0.375)))) / (b ^ 7.0))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(c * c), $MachinePrecision] * N[(N[(N[(N[(c * c), $MachinePrecision] * N[(-1.0546875 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * -0.5625), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

5. Taylor expanded in a around 0

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

7. Simplified 97.4%

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

8. Taylor expanded in c around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left({c}^{2} \cdot \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left({c}^{2}\right), \color{blue}{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(c \cdot c\right), \left(\color{blue}{c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(\color{blue}{c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{\_.f64}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)\right), \color{blue}{\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right)\right) \]

10. Simplified 97.4%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot \left(\frac{-1.0546875 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{7}} + \frac{-0.5625 \cdot a}{{b}^{5}}\right) - \frac{0.375}{b \cdot \left(b \cdot b\right)}\right)\right)} \]

11. Taylor expanded in b around 0

    \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \color{blue}{\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)}{{b}^{7}}\right)}\right)\right)\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\left(\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)\right), \color{blue}{\left({b}^{7}\right)}\right)\right)\right)\right) \]

13. Simplified 97.4%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\frac{\left(-1.0546875 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot -0.5625 + -0.375 \cdot \left(b \cdot b\right)\right)}{{b}^{7}}}\right) \]

14. Final simplification 97.4%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \frac{\left(c \cdot c\right) \cdot \left(-1.0546875 \cdot \left(a \cdot a\right)\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot -0.5625 + \left(b \cdot b\right) \cdot -0.375\right)}{{b}^{7}}\right) \]
15. Add Preprocessing

Alternative 2: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-0.5}{b} + a \cdot \frac{c}{\frac{{b}^{7}}{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -1.0546875\right) + \left(b \cdot b\right) \cdot \left(c \cdot \left(a \cdot -0.5625\right) + b \cdot \left(b \cdot -0.375\right)\right)}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (/ -0.5 b)
   (*
    a
    (/
     c
     (/
      (pow b 7.0)
      (+
       (* (* a a) (* (* c c) -1.0546875))
       (* (* b b) (+ (* c (* a -0.5625)) (* b (* b -0.375)))))))))))

C

double code(double a, double b, double c) {
	return c * ((-0.5 / b) + (a * (c / (pow(b, 7.0) / (((a * a) * ((c * c) * -1.0546875)) + ((b * b) * ((c * (a * -0.5625)) + (b * (b * -0.375)))))))));
}

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) + (a * (c / ((b ** 7.0d0) / (((a * a) * ((c * c) * (-1.0546875d0))) + ((b * b) * ((c * (a * (-0.5625d0))) + (b * (b * (-0.375d0))))))))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-0.5 / b) + (a * (c / (Math.pow(b, 7.0) / (((a * a) * ((c * c) * -1.0546875)) + ((b * b) * ((c * (a * -0.5625)) + (b * (b * -0.375)))))))));
}

Python

def code(a, b, c):
	return c * ((-0.5 / b) + (a * (c / (math.pow(b, 7.0) / (((a * a) * ((c * c) * -1.0546875)) + ((b * b) * ((c * (a * -0.5625)) + (b * (b * -0.375)))))))))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-0.5 / b) + Float64(a * Float64(c / Float64((b ^ 7.0) / Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * -1.0546875)) + Float64(Float64(b * b) * Float64(Float64(c * Float64(a * -0.5625)) + Float64(b * Float64(b * -0.375))))))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-0.5 / b) + (a * (c / ((b ^ 7.0) / (((a * a) * ((c * c) * -1.0546875)) + ((b * b) * ((c * (a * -0.5625)) + (b * (b * -0.375)))))))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-0.5 / b), $MachinePrecision] + N[(a * N[(c / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * -1.0546875), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(N[(c * N[(a * -0.5625), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

5. Taylor expanded in a around 0

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

7. Simplified 97.4%

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

8. Taylor expanded in c around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left({c}^{2} \cdot \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left({c}^{2}\right), \color{blue}{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(c \cdot c\right), \left(\color{blue}{c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(\color{blue}{c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{\_.f64}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)\right), \color{blue}{\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right)\right) \]

10. Simplified 97.4%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot \left(\frac{-1.0546875 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{7}} + \frac{-0.5625 \cdot a}{{b}^{5}}\right) - \frac{0.375}{b \cdot \left(b \cdot b\right)}\right)\right)} \]

11. Taylor expanded in b around 0

    \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \color{blue}{\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)}{{b}^{7}}\right)}\right)\right)\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\left(\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)\right), \color{blue}{\left({b}^{7}\right)}\right)\right)\right)\right) \]

13. Simplified 97.4%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\frac{\left(-1.0546875 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot -0.5625 + -0.375 \cdot \left(b \cdot b\right)\right)}{{b}^{7}}}\right) \]
14. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto c \cdot \frac{\frac{-1}{2}}{b} + \color{blue}{a} \cdot \left(\left(c \cdot c\right) \cdot \frac{\left(\frac{-135}{128} \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot \frac{-9}{16} + \frac{-3}{8} \cdot \left(b \cdot b\right)\right)}{{b}^{7}}\right) \]

    2. *-commutative N/A

       \[\leadsto c \cdot \frac{\frac{-1}{2}}{b} + \left(\left(c \cdot c\right) \cdot \frac{\left(\frac{-135}{128} \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot \frac{-9}{16} + \frac{-3}{8} \cdot \left(b \cdot b\right)\right)}{{b}^{7}}\right) \cdot \color{blue}{a} \]

    3. associate-*l* N/A

       \[\leadsto c \cdot \frac{\frac{-1}{2}}{b} + \left(c \cdot \left(c \cdot \frac{\left(\frac{-135}{128} \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot \frac{-9}{16} + \frac{-3}{8} \cdot \left(b \cdot b\right)\right)}{{b}^{7}}\right)\right) \cdot a \]

    4. associate-*l* N/A

       \[\leadsto c \cdot \frac{\frac{-1}{2}}{b} + c \cdot \color{blue}{\left(\left(c \cdot \frac{\left(\frac{-135}{128} \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot \frac{-9}{16} + \frac{-3}{8} \cdot \left(b \cdot b\right)\right)}{{b}^{7}}\right) \cdot a\right)} \]

    5. distribute-lft-out N/A

       \[\leadsto c \cdot \color{blue}{\left(\frac{\frac{-1}{2}}{b} + \left(c \cdot \frac{\left(\frac{-135}{128} \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot \frac{-9}{16} + \frac{-3}{8} \cdot \left(b \cdot b\right)\right)}{{b}^{7}}\right) \cdot a\right)} \]

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

       \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{-1}{2}}{b} + \left(c \cdot \frac{\left(\frac{-135}{128} \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot \frac{-9}{16} + \frac{-3}{8} \cdot \left(b \cdot b\right)\right)}{{b}^{7}}\right) \cdot a\right)}\right) \]

15. Applied egg-rr 97.0%

    \[\leadsto \color{blue}{c \cdot \left(\frac{-0.5}{b} + \frac{c}{\frac{{b}^{7}}{\left(a \cdot a\right) \cdot \left(-1.0546875 \cdot \left(c \cdot c\right)\right) + \left(b \cdot b\right) \cdot \left(c \cdot \left(a \cdot -0.5625\right) + b \cdot \left(b \cdot -0.375\right)\right)}} \cdot a\right)} \]

16. Final simplification 97.0%

    \[\leadsto c \cdot \left(\frac{-0.5}{b} + a \cdot \frac{c}{\frac{{b}^{7}}{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -1.0546875\right) + \left(b \cdot b\right) \cdot \left(c \cdot \left(a \cdot -0.5625\right) + b \cdot \left(b \cdot -0.375\right)\right)}}\right) \]
17. Add Preprocessing

Alternative 3: 96.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \frac{-0.375 + \frac{\left(c \cdot a\right) \cdot -0.5625}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (/ (* c -0.5) b)
  (*
   a
   (* (* c c) (/ (+ -0.375 (/ (* (* c a) -0.5625) (* b b))) (* b (* b b)))))))

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((c * c) * ((-0.375 + (((c * a) * -0.5625) / (b * b))) / (b * (b * b)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (-0.5d0)) / b) + (a * ((c * c) * (((-0.375d0) + (((c * a) * (-0.5625d0)) / (b * b))) / (b * (b * b)))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((c * c) * ((-0.375 + (((c * a) * -0.5625) / (b * b))) / (b * (b * b)))));
}

Python

def code(a, b, c):
	return ((c * -0.5) / b) + (a * ((c * c) * ((-0.375 + (((c * a) * -0.5625) / (b * b))) / (b * (b * b)))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(c * c) * Float64(Float64(-0.375 + Float64(Float64(Float64(c * a) * -0.5625) / Float64(b * b))) / Float64(b * Float64(b * b))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + (a * ((c * c) * ((-0.375 + (((c * a) * -0.5625) / (b * b))) / (b * (b * b)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(c * c), $MachinePrecision] * N[(N[(-0.375 + N[(N[(N[(c * a), $MachinePrecision] * -0.5625), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

5. Taylor expanded in a around 0

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

7. Simplified 97.4%

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

8. Taylor expanded in c around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left({c}^{2} \cdot \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left({c}^{2}\right), \color{blue}{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(c \cdot c\right), \left(\color{blue}{c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(\color{blue}{c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{\_.f64}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)\right), \color{blue}{\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right)\right) \]

10. Simplified 97.4%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot \left(\frac{-1.0546875 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{7}} + \frac{-0.5625 \cdot a}{{b}^{5}}\right) - \frac{0.375}{b \cdot \left(b \cdot b\right)}\right)\right)} \]

11. Taylor expanded in b around inf

    \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{3}{8}}{{b}^{3}}\right)}\right)\right)\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{3}{8}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right)\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}} + \frac{-3}{8}\right), \left({b}^{3}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot c}{{b}^{2}}\right), \frac{-3}{8}\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right)\right) \]

    5. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot c\right)}{{b}^{2}}\right), \frac{-3}{8}\right), \left({b}^{3}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot c\right)\right), \left({b}^{2}\right)\right), \frac{-3}{8}\right), \left({b}^{3}\right)\right)\right)\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(a \cdot c\right) \cdot \frac{-9}{16}\right), \left({b}^{2}\right)\right), \frac{-3}{8}\right), \left({b}^{3}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a \cdot c\right), \frac{-9}{16}\right), \left({b}^{2}\right)\right), \frac{-3}{8}\right), \left({b}^{3}\right)\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c \cdot a\right), \frac{-9}{16}\right), \left({b}^{2}\right)\right), \frac{-3}{8}\right), \left({b}^{3}\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-9}{16}\right), \left({b}^{2}\right)\right), \frac{-3}{8}\right), \left({b}^{3}\right)\right)\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-9}{16}\right), \left(b \cdot b\right)\right), \frac{-3}{8}\right), \left({b}^{3}\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-9}{16}\right), \mathsf{*.f64}\left(b, b\right)\right), \frac{-3}{8}\right), \left({b}^{3}\right)\right)\right)\right)\right) \]

    13. cube-mult N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-9}{16}\right), \mathsf{*.f64}\left(b, b\right)\right), \frac{-3}{8}\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right)\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-9}{16}\right), \mathsf{*.f64}\left(b, b\right)\right), \frac{-3}{8}\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-9}{16}\right), \mathsf{*.f64}\left(b, b\right)\right), \frac{-3}{8}\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right)\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-9}{16}\right), \mathsf{*.f64}\left(b, b\right)\right), \frac{-3}{8}\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right)\right) \]

    17. *-lowering-*.f64 96.4%

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), \frac{-9}{16}\right), \mathsf{*.f64}\left(b, b\right)\right), \frac{-3}{8}\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right)\right) \]

13. Simplified 96.4%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\frac{\frac{\left(c \cdot a\right) \cdot -0.5625}{b \cdot b} + -0.375}{b \cdot \left(b \cdot b\right)}}\right) \]

14. Final simplification 96.4%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \frac{-0.375 + \frac{\left(c \cdot a\right) \cdot -0.5625}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right) \]
15. Add Preprocessing

Alternative 4: 95.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + ((-0.375 * (c * (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 * (-0.5d0)) / b) + (((-0.375d0) * (c * (c * a))) / (b * (b * b)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

5. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. associate-*r/ N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. associate-/l* N/A

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

   13. *-commutative N/A

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

   14. associate-*r/ N/A

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

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

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

7. Simplified 94.7%

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

Alternative 5: 95.3% accurate, 6.8× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * -0.5) + ((-0.375 * (c * (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 * (-0.5d0)) + (((-0.375d0) * (c * (c * a))) / (b * b))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

5. Taylor expanded in b around inf

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

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

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

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

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

   3. *-commutative N/A

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

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

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

   5. associate-*r/ N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 94.7%

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

7. Simplified 94.7%

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

Alternative 6: 95.0% accurate, 6.8× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c / ((a / -0.3333333333333333) * ((c / (b / -0.5)) + ((b * 0.6666666666666666) / 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 = c / ((a / (-0.3333333333333333d0)) * ((c / (b / (-0.5d0))) + ((b * 0.6666666666666666d0) / a)))
end function

Java

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

Python

def code(a, b, c):
	return c / ((a / -0.3333333333333333) * ((c / (b / -0.5)) + ((b * 0.6666666666666666) / a)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = c / ((a / -0.3333333333333333) * ((c / (b / -0.5)) + ((b * 0.6666666666666666) / a)));
end

Wolfram

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

Derivation

1. Initial program 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

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

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

   9. associate-/r* N/A

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

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

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

   11. metadata-eval N/A

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

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

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

6. Applied egg-rr 17.8%

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

7. Taylor expanded in c around 0

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

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

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

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

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

   3. associate-*r/ N/A

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

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

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

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

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

   6. associate-*r/ N/A

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

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

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 94.3%

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

9. Simplified 94.3%

   \[\leadsto \frac{\frac{0.3333333333333333}{a}}{\color{blue}{\frac{\frac{-0.6666666666666666 \cdot b}{a} + \frac{c \cdot 0.5}{b}}{c}}} \]
10. Step-by-step derivation

    1. div-inv N/A

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

    2. metadata-eval N/A

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

    3. distribute-neg-frac N/A

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

    4. clear-num N/A

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

    5. distribute-neg-frac N/A

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

    6. metadata-eval N/A

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

    7. clear-num N/A

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

    8. frac-2neg N/A

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

    9. frac-times N/A

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

    10. neg-mul-1 N/A

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

    11. remove-double-neg N/A

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

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

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

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

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

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

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

    15. +-commutative N/A

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

    16. distribute-neg-in N/A

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

11. Applied egg-rr 94.4%

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

Alternative 7: 95.0% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

5. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

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

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

   7. +-commutative N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

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

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

   14. associate-*r/ N/A

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

   15. associate-*l/ N/A

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

   16. associate-*r* N/A

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

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

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

7. Simplified 94.3%

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

8. Final simplification 94.3%

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

Alternative 8: 94.8% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.3333333333333333d0 * (c / (a * (((-0.6666666666666666d0) / (a / b)) + (c / (b / 0.5d0)))))
end function

Java

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

Python

def code(a, b, c):
	return 0.3333333333333333 * (c / (a * ((-0.6666666666666666 / (a / b)) + (c / (b / 0.5)))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = 0.3333333333333333 * (c / (a * ((-0.6666666666666666 / (a / b)) + (c / (b / 0.5)))));
end

Wolfram

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

Derivation

1. Initial program 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

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

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

   9. associate-/r* N/A

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

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

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

   11. metadata-eval N/A

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

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

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

6. Applied egg-rr 17.8%

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

7. Taylor expanded in c around 0

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

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

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

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

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

   3. associate-*r/ N/A

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

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

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

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

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

   6. associate-*r/ N/A

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

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

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 94.3%

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

9. Simplified 94.3%

   \[\leadsto \frac{\frac{0.3333333333333333}{a}}{\color{blue}{\frac{\frac{-0.6666666666666666 \cdot b}{a} + \frac{c \cdot 0.5}{b}}{c}}} \]
10. Step-by-step derivation

    1. div-inv N/A

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

    2. associate-/l* N/A

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

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

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

    4. div-inv N/A

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

    5. clear-num N/A

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

    6. frac-times N/A

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

    7. metadata-eval N/A

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

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

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

    9. neg-mul-1 N/A

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

    10. remove-double-neg N/A

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

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

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

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

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

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

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

    14. associate-/l* N/A

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

    15. clear-num N/A

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

    16. un-div-inv N/A

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

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

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

    18. /-lowering-/.f64 N/A

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

    19. associate-/l* N/A

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

    20. clear-num N/A

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

    21. un-div-inv N/A

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

11. Applied egg-rr 94.2%

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

Alternative 9: 94.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.3333333333333333}{a}}{\frac{0.5}{b} + -0.6666666666666666 \cdot \frac{b}{c \cdot a}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return (0.3333333333333333 / a) / ((0.5 / b) + (-0.6666666666666666 * (b / (c * 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 = (0.3333333333333333d0 / a) / ((0.5d0 / b) + ((-0.6666666666666666d0) * (b / (c * a))))
end function

Java

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

Python

def code(a, b, c):
	return (0.3333333333333333 / a) / ((0.5 / b) + (-0.6666666666666666 * (b / (c * a))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (0.3333333333333333 / a) / ((0.5 / b) + (-0.6666666666666666 * (b / (c * a))));
end

Wolfram

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

Derivation

1. Initial program 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

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

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

   9. associate-/r* N/A

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

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

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

   11. metadata-eval N/A

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

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

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

6. Applied egg-rr 17.8%

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

7. Taylor expanded in c around 0

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

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

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

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

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

   3. associate-*r/ N/A

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

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

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

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

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

   6. associate-*r/ N/A

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

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

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 94.3%

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

9. Simplified 94.3%

   \[\leadsto \frac{\frac{0.3333333333333333}{a}}{\color{blue}{\frac{\frac{-0.6666666666666666 \cdot b}{a} + \frac{c \cdot 0.5}{b}}{c}}} \]

10. Taylor expanded in a around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, a\right), \color{blue}{\left(\frac{-2}{3} \cdot \frac{b}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b}\right)}\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

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

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

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

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

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

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

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

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

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

    8. *-commutative N/A

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

    9. *-lowering-*.f64 94.2%

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

12. Simplified 94.2%

    \[\leadsto \frac{\frac{0.3333333333333333}{a}}{\color{blue}{\frac{0.5}{b} + -0.6666666666666666 \cdot \frac{b}{c \cdot a}}} \]
13. Add Preprocessing

Alternative 10: 90.8% 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 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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 90.2%

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

7. Simplified 90.2%

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

Alternative 11: 90.5% 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 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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 90.2%

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

7. Simplified 90.2%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
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 89.9%

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

9. Applied egg-rr 89.9%

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

10. Final simplification 89.9%

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

Alternative 12: 3.3% 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 17.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 17.8%

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

3. Simplified 17.8%

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

6. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{\frac{\frac{b}{\frac{0.3333333333333333}{a}} + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \left(a \cdot -3\right)}{a \cdot \left(a \cdot -9\right)}} \]

7. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. distribute-rgt-out N/A

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

   3. metadata-eval N/A

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

   4. mul0-rgt N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 3.3%

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

9. Simplified 3.3%

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

    1. div0 3.3%

       \[\leadsto 0 \]

11. Applied egg-rr 3.3%

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

Reproduce

herbie shell --seed 2024159 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))