Cubic critical, wide range

Percentage Accurate: 18.1% → 97.9%

Time: 15.3s

Alternatives: 8

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: 18.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot {\left(\frac{c}{b}\right)}^{2} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -0.5 c)
   (*
    a
    (+
     (* -0.375 (pow (/ c b) 2.0))
     (*
      a
      (+
       (* -1.0546875 (/ (* a (pow c 4.0)) (pow b 6.0)))
       (* -0.5625 (/ (pow c 3.0) (pow b 4.0))))))))
  b))

C

double code(double a, double b, double c) {
	return ((-0.5 * c) + (a * ((-0.375 * pow((c / b), 2.0)) + (a * ((-1.0546875 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-0.5625 * (pow(c, 3.0) / pow(b, 4.0)))))))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-0.5d0) * c) + (a * (((-0.375d0) * ((c / b) ** 2.0d0)) + (a * (((-1.0546875d0) * ((a * (c ** 4.0d0)) / (b ** 6.0d0))) + ((-0.5625d0) * ((c ** 3.0d0) / (b ** 4.0d0)))))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((-0.5 * c) + (a * ((-0.375 * Math.pow((c / b), 2.0)) + (a * ((-1.0546875 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 6.0))) + (-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 4.0)))))))) / b;
}

Python

def code(a, b, c):
	return ((-0.5 * c) + (a * ((-0.375 * math.pow((c / b), 2.0)) + (a * ((-1.0546875 * ((a * math.pow(c, 4.0)) / math.pow(b, 6.0))) + (-0.5625 * (math.pow(c, 3.0) / math.pow(b, 4.0)))))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-0.5 * c) + Float64(a * Float64(Float64(-0.375 * (Float64(c / b) ^ 2.0)) + Float64(a * Float64(Float64(-1.0546875 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 6.0))) + Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 4.0)))))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-0.5 * c) + (a * ((-0.375 * ((c / b) ^ 2.0)) + (a * ((-1.0546875 * ((a * (c ^ 4.0)) / (b ^ 6.0))) + (-0.5625 * ((c ^ 3.0) / (b ^ 4.0)))))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.5 * c), $MachinePrecision] + N[(a * N[(N[(-0.375 * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-1.0546875 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in b around inf 98.1%

   \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
4. Step-by-step derivation

   1. +-commutative 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(\left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}}{b} \]

   2. *-un-lft-identity 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\color{blue}{1 \cdot \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)} + -0.5 \cdot c\right)}{b} \]

   3. fma-define 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{fma}\left(1, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}, -0.5 \cdot c\right)}}{b} \]

5. Applied egg-rr 98.1%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right), c \cdot -0.5\right)}}{b} \]
6. Step-by-step derivation

   1. fma-undefine 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(1 \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}}{b} \]

   2. *-lft-identity 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\color{blue}{\mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)} + c \cdot -0.5\right)}{b} \]

   3. unpow2 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   4. unpow2 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   5. times-frac 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   6. unpow1 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right), \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   7. pow-plus 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   8. metadata-eval 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   9. associate-/l* 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, \color{blue}{-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{6}}}\right) + c \cdot -0.5\right)}{b} \]

   10. associate-/l* 98.1%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, -0.16666666666666666 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)}\right) + c \cdot -0.5\right)}{b} \]

   11. *-commutative 98.1%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, -0.16666666666666666 \cdot \left({\color{blue}{\left(c \cdot a\right)}}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right) + c \cdot -0.5\right)}{b} \]

   12. *-commutative 98.1%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, -0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right) + \color{blue}{-0.5 \cdot c}\right)}{b} \]

7. Simplified 98.1%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, -0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right) + -0.5 \cdot c\right)}}{b} \]

8. Taylor expanded in a around 0 98.1%

   \[\leadsto \frac{\color{blue}{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}}{b} \]
9. Step-by-step derivation

   1. associate-*r/ 98.1%

      \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(\color{blue}{\frac{-0.375 \cdot {c}^{2}}{{b}^{2}}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

10. Applied egg-rr 98.1%

    \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(\color{blue}{\frac{-0.375 \cdot {c}^{2}}{{b}^{2}}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]
11. Step-by-step derivation

    1. associate-*r/ 98.1%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(\color{blue}{-0.375 \cdot \frac{{c}^{2}}{{b}^{2}}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    2. *-rgt-identity 98.1%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \frac{\color{blue}{{c}^{2} \cdot 1}}{{b}^{2}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    3. associate-/l* 98.1%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{2}}\right)} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    4. unpow2 98.1%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot \frac{1}{{b}^{2}}\right) + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    5. unpow2 98.1%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{1}{\color{blue}{b \cdot b}}\right) + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    6. associate-/r* 98.1%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\frac{\frac{1}{b}}{b}}\right) + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    7. *-rgt-identity 98.1%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \left(\left(c \cdot c\right) \cdot \frac{\color{blue}{\frac{1}{b} \cdot 1}}{b}\right) + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    8. associate-*r/ 98.1%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(\frac{1}{b} \cdot \frac{1}{b}\right)}\right) + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    9. swap-sqr 98.1%

       \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \color{blue}{\left(\left(c \cdot \frac{1}{b}\right) \cdot \left(c \cdot \frac{1}{b}\right)\right)} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    10. *-commutative 98.1%

        \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \left(\color{blue}{\left(\frac{1}{b} \cdot c\right)} \cdot \left(c \cdot \frac{1}{b}\right)\right) + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    11. *-commutative 98.1%

        \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \left(\left(\frac{1}{b} \cdot c\right) \cdot \color{blue}{\left(\frac{1}{b} \cdot c\right)}\right) + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    12. unpow1 98.1%

        \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \left(\color{blue}{{\left(\frac{1}{b} \cdot c\right)}^{1}} \cdot \left(\frac{1}{b} \cdot c\right)\right) + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    13. pow-plus 98.1%

        \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot \color{blue}{{\left(\frac{1}{b} \cdot c\right)}^{\left(1 + 1\right)}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    14. associate-*l/ 98.1%

        \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot {\color{blue}{\left(\frac{1 \cdot c}{b}\right)}}^{\left(1 + 1\right)} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    15. *-lft-identity 98.1%

        \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot {\left(\frac{\color{blue}{c}}{b}\right)}^{\left(1 + 1\right)} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

    16. metadata-eval 98.1%

        \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(-0.375 \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]

12. Simplified 98.1%

    \[\leadsto \frac{-0.5 \cdot c + a \cdot \left(\color{blue}{-0.375 \cdot {\left(\frac{c}{b}\right)}^{2}} + a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -0.5625 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]
13. Add Preprocessing

Alternative 2: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-1.0546875 \cdot \frac{c \cdot {a}^{3}}{{b}^{6}} + -0.5625 \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (*
     c
     (+
      (* -0.375 (/ a (pow b 2.0)))
      (*
       c
       (+
        (* -1.0546875 (/ (* c (pow a 3.0)) (pow b 6.0)))
        (* -0.5625 (/ (pow a 2.0) (pow b 4.0)))))))
    0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * ((-0.375 * (a / pow(b, 2.0))) + (c * ((-1.0546875 * ((c * pow(a, 3.0)) / pow(b, 6.0))) + (-0.5625 * (pow(a, 2.0) / pow(b, 4.0))))))) - 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 * ((c * (((-0.375d0) * (a / (b ** 2.0d0))) + (c * (((-1.0546875d0) * ((c * (a ** 3.0d0)) / (b ** 6.0d0))) + ((-0.5625d0) * ((a ** 2.0d0) / (b ** 4.0d0))))))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((c * ((-0.375 * (a / Math.pow(b, 2.0))) + (c * ((-1.0546875 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 6.0))) + (-0.5625 * (Math.pow(a, 2.0) / Math.pow(b, 4.0))))))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((c * ((-0.375 * (a / math.pow(b, 2.0))) + (c * ((-1.0546875 * ((c * math.pow(a, 3.0)) / math.pow(b, 6.0))) + (-0.5625 * (math.pow(a, 2.0) / math.pow(b, 4.0))))))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * Float64(Float64(-0.375 * Float64(a / (b ^ 2.0))) + Float64(c * Float64(Float64(-1.0546875 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 6.0))) + Float64(-0.5625 * Float64((a ^ 2.0) / (b ^ 4.0))))))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((c * ((-0.375 * (a / (b ^ 2.0))) + (c * ((-1.0546875 * ((c * (a ^ 3.0)) / (b ^ 6.0))) + (-0.5625 * ((a ^ 2.0) / (b ^ 4.0))))))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-1.0546875 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in b around inf 98.1%

   \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
4. Step-by-step derivation

   1. +-commutative 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(\left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right) + -0.5 \cdot c\right)}}{b} \]

   2. *-un-lft-identity 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\color{blue}{1 \cdot \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)} + -0.5 \cdot c\right)}{b} \]

   3. fma-define 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{fma}\left(1, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}, -0.5 \cdot c\right)}}{b} \]

5. Applied egg-rr 98.1%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right), c \cdot -0.5\right)}}{b} \]
6. Step-by-step derivation

   1. fma-undefine 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(1 \cdot \mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}}{b} \]

   2. *-lft-identity 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\color{blue}{\mathsf{fma}\left(-0.375, a \cdot \frac{{c}^{2}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right)} + c \cdot -0.5\right)}{b} \]

   3. unpow2 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   4. unpow2 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   5. times-frac 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   6. unpow1 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right), \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   7. pow-plus 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{\left(1 + 1\right)}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   8. metadata-eval 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}, \frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 6.328125\right)}{a \cdot {b}^{6}}\right) + c \cdot -0.5\right)}{b} \]

   9. associate-/l* 98.1%

      \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, \color{blue}{-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{a \cdot {b}^{6}}}\right) + c \cdot -0.5\right)}{b} \]

   10. associate-/l* 98.1%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, -0.16666666666666666 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)}\right) + c \cdot -0.5\right)}{b} \]

   11. *-commutative 98.1%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, -0.16666666666666666 \cdot \left({\color{blue}{\left(c \cdot a\right)}}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right) + c \cdot -0.5\right)}{b} \]

   12. *-commutative 98.1%

       \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, -0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right) + \color{blue}{-0.5 \cdot c}\right)}{b} \]

7. Simplified 98.1%

   \[\leadsto \frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \color{blue}{\left(\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, -0.16666666666666666 \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{a \cdot {b}^{6}}\right)\right) + -0.5 \cdot c\right)}}{b} \]

8. Taylor expanded in c around 0 98.1%

   \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -0.5625 \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - 0.5\right)}}{b} \]

9. Final simplification 98.1%

   \[\leadsto \frac{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{2}} + c \cdot \left(-1.0546875 \cdot \frac{c \cdot {a}^{3}}{{b}^{6}} + -0.5625 \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - 0.5\right)}{b} \]
10. Add Preprocessing

Alternative 3: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (*
   a
   (+
    (* -0.5625 (/ (* a (pow c 3.0)) (pow b 5.0)))
    (* -0.375 (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * pow(c, 3.0)) / pow(b, 5.0))) + (-0.375 * (pow(c, 2.0) / pow(b, 3.0)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + (a * (((-0.5625d0) * ((a * (c ** 3.0d0)) / (b ** 5.0d0))) + ((-0.375d0) * ((c ** 2.0d0) / (b ** 3.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (-0.375 * (Math.pow(c, 2.0) / Math.pow(b, 3.0)))));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (-0.375 * (math.pow(c, 2.0) / math.pow(b, 3.0)))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(a * Float64(Float64(-0.5625 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (a * ((-0.5625 * ((a * (c ^ 3.0)) / (b ^ 5.0))) + (-0.375 * ((c ^ 2.0) / (b ^ 3.0)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in a around 0 97.5%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + a \cdot \left(-0.5625 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
4. Add Preprocessing

Alternative 4: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{c \cdot a}{{b}^{4}} + 0.375 \cdot \left(\frac{1}{b} \cdot \frac{-1}{b}\right)\right)\right) - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (*
     c
     (*
      a
      (+
       (* -0.5625 (/ (* c a) (pow b 4.0)))
       (* 0.375 (* (/ 1.0 b) (/ -1.0 b))))))
    0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * (a * ((-0.5625 * ((c * a) / pow(b, 4.0))) + (0.375 * ((1.0 / b) * (-1.0 / b)))))) - 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 * ((c * (a * (((-0.5625d0) * ((c * a) / (b ** 4.0d0))) + (0.375d0 * ((1.0d0 / b) * ((-1.0d0) / b)))))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((c * (a * ((-0.5625 * ((c * a) / Math.pow(b, 4.0))) + (0.375 * ((1.0 / b) * (-1.0 / b)))))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((c * (a * ((-0.5625 * ((c * a) / math.pow(b, 4.0))) + (0.375 * ((1.0 / b) * (-1.0 / b)))))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(-0.5625 * Float64(Float64(c * a) / (b ^ 4.0))) + Float64(0.375 * Float64(Float64(1.0 / b) * Float64(-1.0 / b)))))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((c * (a * ((-0.5625 * ((c * a) / (b ^ 4.0))) + (0.375 * ((1.0 / b) * (-1.0 / b)))))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(a * N[(N[(-0.5625 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.375 * N[(N[(1.0 / b), $MachinePrecision] * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in b around inf 98.1%

   \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Taylor expanded in c around 0 97.4%

   \[\leadsto \frac{\color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}}{b} \]

5. Taylor expanded in a around 0 97.4%

   \[\leadsto \frac{c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot \frac{1}{{b}^{2}}\right)\right)} - 0.5\right)}{b} \]
6. Step-by-step derivation

   1. inv-pow 97.4%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot \color{blue}{{\left({b}^{2}\right)}^{-1}}\right)\right) - 0.5\right)}{b} \]

   2. unpow2 97.4%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot {\color{blue}{\left(b \cdot b\right)}}^{-1}\right)\right) - 0.5\right)}{b} \]

   3. metadata-eval 97.4%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot {\left(b \cdot b\right)}^{\color{blue}{\left(-1\right)}}\right)\right) - 0.5\right)}{b} \]

   4. unpow-prod-down 97.4%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot \color{blue}{\left({b}^{\left(-1\right)} \cdot {b}^{\left(-1\right)}\right)}\right)\right) - 0.5\right)}{b} \]

   5. metadata-eval 97.4%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot \left({b}^{\color{blue}{-1}} \cdot {b}^{\left(-1\right)}\right)\right)\right) - 0.5\right)}{b} \]

   6. inv-pow 97.4%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot \left(\color{blue}{\frac{1}{b}} \cdot {b}^{\left(-1\right)}\right)\right)\right) - 0.5\right)}{b} \]

   7. metadata-eval 97.4%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot \left(\frac{1}{b} \cdot {b}^{\color{blue}{-1}}\right)\right)\right) - 0.5\right)}{b} \]

   8. inv-pow 97.4%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot \left(\frac{1}{b} \cdot \color{blue}{\frac{1}{b}}\right)\right)\right) - 0.5\right)}{b} \]

7. Applied egg-rr 97.4%

   \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{4}} - 0.375 \cdot \color{blue}{\left(\frac{1}{b} \cdot \frac{1}{b}\right)}\right)\right) - 0.5\right)}{b} \]

8. Final simplification 97.4%

   \[\leadsto \frac{c \cdot \left(c \cdot \left(a \cdot \left(-0.5625 \cdot \frac{c \cdot a}{{b}^{4}} + 0.375 \cdot \left(\frac{1}{b} \cdot \frac{-1}{b}\right)\right)\right) - 0.5\right)}{b} \]
9. Add Preprocessing

Alternative 5: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c (- (* -0.375 (/ (* c a) (pow b 2.0))) 0.5)) b))

C

double code(double a, double b, double c) {
	return (c * ((-0.375 * ((c * a) / pow(b, 2.0))) - 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 * (((-0.375d0) * ((c * a) / (b ** 2.0d0))) - 0.5d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((-0.375 * ((c * a) / Math.pow(b, 2.0))) - 0.5)) / b;
}

Python

def code(a, b, c):
	return (c * ((-0.375 * ((c * a) / math.pow(b, 2.0))) - 0.5)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(c * a) / (b ^ 2.0))) - 0.5)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((-0.375 * ((c * a) / (b ^ 2.0))) - 0.5)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in b around inf 98.1%

   \[\leadsto \color{blue}{\frac{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-0.5 \cdot c + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Taylor expanded in c around 0 96.2%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}}{b} \]

5. Final simplification 96.2%

   \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5\right)}{b} \]
6. Add Preprocessing

Alternative 6: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (- (* -0.375 (/ (* c a) (pow b 2.0))) 0.5) b)))

C

double code(double a, double b, double c) {
	return c * (((-0.375 * ((c * a) / pow(b, 2.0))) - 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 * ((((-0.375d0) * ((c * a) / (b ** 2.0d0))) - 0.5d0) / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * (((-0.375 * ((c * a) / Math.pow(b, 2.0))) - 0.5) / b);
}

Python

def code(a, b, c):
	return c * (((-0.375 * ((c * a) / math.pow(b, 2.0))) - 0.5) / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(Float64(-0.375 * Float64(Float64(c * a) / (b ^ 2.0))) - 0.5) / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * (((-0.375 * ((c * a) / (b ^ 2.0))) - 0.5) / b);
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in c around 0 97.2%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]

4. Taylor expanded in b around inf 95.9%

   \[\leadsto c \cdot \color{blue}{\frac{-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5}{b}} \]

5. Final simplification 95.9%

   \[\leadsto c \cdot \frac{-0.375 \cdot \frac{c \cdot a}{{b}^{2}} - 0.5}{b} \]
6. Add Preprocessing

Alternative 7: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}}\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (* c (* -0.375 (/ a (pow b 3.0)))) (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((c * (-0.375 * (a / pow(b, 3.0)))) - (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 * ((c * ((-0.375d0) * (a / (b ** 3.0d0)))) - (0.5d0 / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * (-0.375 * (a / Math.pow(b, 3.0)))) - (0.5 / b));
}

Python

def code(a, b, c):
	return c * ((c * (-0.375 * (a / math.pow(b, 3.0)))) - (0.5 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(-0.375 * Float64(a / (b ^ 3.0)))) - Float64(0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * (-0.375 * (a / (b ^ 3.0)))) - (0.5 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in c around 0 97.2%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) - 0.5 \cdot \frac{1}{b}\right)} \]

4. Taylor expanded in c around 0 95.9%

   \[\leadsto c \cdot \left(\color{blue}{-0.375 \cdot \frac{a \cdot c}{{b}^{3}}} - 0.5 \cdot \frac{1}{b}\right) \]
5. Step-by-step derivation

   1. associate-*r/ 95.9%

      \[\leadsto c \cdot \left(\color{blue}{\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}}} - 0.5 \cdot \frac{1}{b}\right) \]

   2. associate-*r* 95.9%

      \[\leadsto c \cdot \left(\frac{\color{blue}{\left(-0.375 \cdot a\right) \cdot c}}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \]

   3. associate-*l/ 95.9%

      \[\leadsto c \cdot \left(\color{blue}{\frac{-0.375 \cdot a}{{b}^{3}} \cdot c} - 0.5 \cdot \frac{1}{b}\right) \]

   4. *-commutative 95.9%

      \[\leadsto c \cdot \left(\color{blue}{c \cdot \frac{-0.375 \cdot a}{{b}^{3}}} - 0.5 \cdot \frac{1}{b}\right) \]

   5. associate-*r/ 95.9%

      \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(-0.375 \cdot \frac{a}{{b}^{3}}\right)} - 0.5 \cdot \frac{1}{b}\right) \]

6. Simplified 95.9%

   \[\leadsto c \cdot \left(\color{blue}{c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}}\right)} - 0.5 \cdot \frac{1}{b}\right) \]

7. Taylor expanded in b around 0 95.9%

   \[\leadsto c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}}\right) - \color{blue}{\frac{0.5}{b}}\right) \]
8. Add Preprocessing

Alternative 8: 90.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in b around inf 91.5%

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

Reproduce

herbie shell --seed 2024139 
(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)))