Cubic critical, wide range

Percentage Accurate: 17.7% → 97.4%

Time: 16.0s

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: 17.7% 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: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, c \cdot a, -1.0546875 \cdot {\left(\frac{c \cdot a}{b}\right)}^{2}\right)}{{b}^{5}} + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (-
   (*
    c
    (*
     a
     (+
      (/
       (fma -0.5625 (* c a) (* -1.0546875 (pow (/ (* c a) b) 2.0)))
       (pow b 5.0))
      (* 0.375 (/ -1.0 (pow b 3.0))))))
   (/ 0.5 b))))

C

double code(double a, double b, double c) {
	return c * ((c * (a * ((fma(-0.5625, (c * a), (-1.0546875 * pow(((c * a) / b), 2.0))) / pow(b, 5.0)) + (0.375 * (-1.0 / pow(b, 3.0)))))) - (0.5 / b));
}

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(a * Float64(Float64(fma(-0.5625, Float64(c * a), Float64(-1.0546875 * (Float64(Float64(c * a) / b) ^ 2.0))) / (b ^ 5.0)) + Float64(0.375 * Float64(-1.0 / (b ^ 3.0)))))) - Float64(0.5 / b)))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(a * N[(N[(N[(-0.5625 * N[(c * a), $MachinePrecision] + N[(-1.0546875 * N[Power[N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.375 * N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in c around 0 97.0%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

5. Simplified 97.0%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

6. Taylor expanded in c around 0 97.0%

   \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]
7. Step-by-step derivation

   1. associate-*r/ 97.0%

      \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{\frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]

8. Simplified 97.0%

   \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{\frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]

9. Taylor expanded in a around 0 97.0%

   \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(-1.0546875 \cdot \frac{a \cdot {c}^{2}}{{b}^{7}} + -0.5625 \cdot \frac{c}{{b}^{5}}\right) - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)} - \frac{0.5}{b}\right) \]

10. Taylor expanded in b around inf 97.0%

    \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\color{blue}{\frac{-1.0546875 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + -0.5625 \cdot \left(a \cdot c\right)}{{b}^{5}}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]
11. Step-by-step derivation

    1. +-commutative 97.0%

       \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\color{blue}{-0.5625 \cdot \left(a \cdot c\right) + -1.0546875 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}}}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    2. fma-define 97.0%

       \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right)}}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    3. associate-/l* 97.0%

       \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}\right)}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    4. unpow2 97.0%

       \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    5. unpow2 97.0%

       \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \left(\left(a \cdot a\right) \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)\right)}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    6. unpow2 97.0%

       \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \left(\left(a \cdot a\right) \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)\right)}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    7. times-frac 97.0%

       \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)\right)}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    8. swap-sqr 97.0%

       \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \color{blue}{\left(\left(a \cdot \frac{c}{b}\right) \cdot \left(a \cdot \frac{c}{b}\right)\right)}\right)}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    9. associate-/l* 97.0%

       \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \left(\color{blue}{\frac{a \cdot c}{b}} \cdot \left(a \cdot \frac{c}{b}\right)\right)\right)}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    10. associate-/l* 97.0%

        \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \left(\frac{a \cdot c}{b} \cdot \color{blue}{\frac{a \cdot c}{b}}\right)\right)}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

    11. unpow2 97.0%

        \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot \color{blue}{{\left(\frac{a \cdot c}{b}\right)}^{2}}\right)}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

12. Simplified 97.0%

    \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\color{blue}{\frac{\mathsf{fma}\left(-0.5625, a \cdot c, -1.0546875 \cdot {\left(\frac{a \cdot c}{b}\right)}^{2}\right)}{{b}^{5}}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]

13. Final simplification 97.0%

    \[\leadsto c \cdot \left(c \cdot \left(a \cdot \left(\frac{\mathsf{fma}\left(-0.5625, c \cdot a, -1.0546875 \cdot {\left(\frac{c \cdot a}{b}\right)}^{2}\right)}{{b}^{5}} + 0.375 \cdot \frac{-1}{{b}^{3}}\right)\right) - \frac{0.5}{b}\right) \]
14. Add Preprocessing

Alternative 2: 97.0% 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 17.1%

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

3. Taylor expanded in a around 0 96.4%

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

4. Final simplification 96.4%

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

Alternative 3: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(c \cdot \left(-0.5625 \cdot \frac{c \cdot {a}^{2}}{{b}^{5}} + -0.375 \cdot \frac{a}{{b}^{3}}\right) + 0.5 \cdot \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (+
     (* -0.5625 (/ (* c (pow a 2.0)) (pow b 5.0)))
     (* -0.375 (/ a (pow b 3.0)))))
   (* 0.5 (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((c * pow(a, 2.0)) / pow(b, 5.0))) + (-0.375 * (a / pow(b, 3.0))))) + (0.5 * (-1.0 / 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 = c * ((c * (((-0.5625d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) + ((-0.375d0) * (a / (b ** 3.0d0))))) + (0.5d0 * ((-1.0d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((c * ((-0.5625 * ((c * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) + (-0.375 * (a / Math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)));
}

Python

def code(a, b, c):
	return c * ((c * ((-0.5625 * ((c * math.pow(a, 2.0)) / math.pow(b, 5.0))) + (-0.375 * (a / math.pow(b, 3.0))))) + (0.5 * (-1.0 / b)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(c * Float64(Float64(-0.5625 * Float64(Float64(c * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.375 * Float64(a / (b ^ 3.0))))) + Float64(0.5 * Float64(-1.0 / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((-0.5625 * ((c * (a ^ 2.0)) / (b ^ 5.0))) + (-0.375 * (a / (b ^ 3.0))))) + (0.5 * (-1.0 / b)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(-0.5625 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in c around 0 96.1%

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

4. Final simplification 96.1%

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

Alternative 4: 96.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((c * (a * ((-0.5625 * (a * (c / pow(b, 5.0)))) - (0.375 / 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 * (a * (((-0.5625d0) * (a * (c / (b ** 5.0d0)))) - (0.375d0 / (b ** 3.0d0))))) - (0.5d0 / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in c around 0 97.0%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

5. Simplified 97.0%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

6. Taylor expanded in c around 0 97.0%

   \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]
7. Step-by-step derivation

   1. associate-*r/ 97.0%

      \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{\frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]

8. Simplified 97.0%

   \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{\frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]

9. Taylor expanded in a around 0 96.1%

   \[\leadsto c \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(-0.5625 \cdot \frac{a \cdot c}{{b}^{5}} - 0.375 \cdot \frac{1}{{b}^{3}}\right)\right)} - \frac{0.5}{b}\right) \]
10. Step-by-step derivation

    1. associate-/l* 96.1%

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

    2. associate-*r/ 96.1%

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

    3. metadata-eval 96.1%

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

11. Simplified 96.1%

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

12. Final simplification 96.1%

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

Alternative 5: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * 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(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in a around 0 95.1%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

4. Final simplification 95.1%

   \[\leadsto -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \]
5. Add Preprocessing

Alternative 6: 95.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return fma(-0.375, (a * pow((c / b), 2.0)), (c * -0.5)) / b;
}

Julia

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

Wolfram

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

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in b around inf 94.4%

   \[\leadsto \frac{\color{blue}{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}}{b}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. associate-/l* 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b}}{3 \cdot a} \]

5. Applied egg-rr 94.4%

   \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b}}{3 \cdot a} \]

   2. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\left(a \cdot a\right) \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)}{b}}{3 \cdot a} \]

   3. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\left(a \cdot a\right) \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)}{b}}{3 \cdot a} \]

   4. times-frac 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)}{b}}{3 \cdot a} \]

   5. swap-sqr 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left(\left(a \cdot \frac{c}{b}\right) \cdot \left(a \cdot \frac{c}{b}\right)\right)}}{b}}{3 \cdot a} \]

   6. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{{\left(a \cdot \frac{c}{b}\right)}^{2}}}{b}}{3 \cdot a} \]

7. Simplified 94.4%

   \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{{\left(a \cdot \frac{c}{b}\right)}^{2}}}{b}}{3 \cdot a} \]

8. Taylor expanded in b around inf 95.1%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
9. Step-by-step derivation

   1. +-commutative 95.1%

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

   2. fma-define 95.1%

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

   3. associate-/l* 95.1%

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

   4. unpow2 95.1%

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

   5. unpow2 95.1%

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

   6. times-frac 95.1%

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

   7. unpow2 95.1%

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

   8. *-commutative 95.1%

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

10. Simplified 95.1%

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

11. Final simplification 95.1%

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

Alternative 7: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in b around inf 95.1%

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

4. Taylor expanded in c around 0 94.8%

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

   1. associate-/l* 94.8%

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

   2. associate-*r/ 94.8%

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

   3. metadata-eval 94.8%

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

6. Simplified 94.8%

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

7. Final simplification 94.8%

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

Alternative 8: 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 17.1%

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

3. Taylor expanded in c around 0 97.0%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
4. Step-by-step derivation

5. Simplified 97.0%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

6. Taylor expanded in c around 0 97.0%

   \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{-1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]
7. Step-by-step derivation

   1. associate-*r/ 97.0%

      \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{\frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]

8. Simplified 97.0%

   \[\leadsto c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, \color{blue}{\frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{7}}}\right)\right) - \frac{0.5}{b}\right) \]

9. Taylor expanded in a around 0 94.8%

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

    1. associate-*r/ 94.8%

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

    2. *-commutative 94.8%

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

11. Simplified 94.8%

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

12. Taylor expanded in b around inf 94.8%

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

13. Final simplification 94.8%

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

Alternative 9: 94.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ 1.0 (/ (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b))) c))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (1.0d0 / ((((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))) / c)) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0);
}

Python

def code(a, b, c):
	return (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b))) / c)) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in b around inf 94.4%

   \[\leadsto \frac{\color{blue}{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}}{b}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. associate-/l* 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b}}{3 \cdot a} \]

5. Applied egg-rr 94.4%

   \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b}}{3 \cdot a} \]

   2. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\left(a \cdot a\right) \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)}{b}}{3 \cdot a} \]

   3. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\left(a \cdot a\right) \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)}{b}}{3 \cdot a} \]

   4. times-frac 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)}{b}}{3 \cdot a} \]

   5. swap-sqr 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left(\left(a \cdot \frac{c}{b}\right) \cdot \left(a \cdot \frac{c}{b}\right)\right)}}{b}}{3 \cdot a} \]

   6. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{{\left(a \cdot \frac{c}{b}\right)}^{2}}}{b}}{3 \cdot a} \]

7. Simplified 94.4%

   \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{{\left(a \cdot \frac{c}{b}\right)}^{2}}}{b}}{3 \cdot a} \]
8. Step-by-step derivation

   1. clear-num 94.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b}{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot {\left(a \cdot \frac{c}{b}\right)}^{2}}}}}{3 \cdot a} \]

   2. inv-pow 94.3%

      \[\leadsto \frac{\color{blue}{{\left(\frac{b}{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot {\left(a \cdot \frac{c}{b}\right)}^{2}}\right)}^{-1}}}{3 \cdot a} \]

   3. +-commutative 94.3%

      \[\leadsto \frac{{\left(\frac{b}{\color{blue}{-1.125 \cdot {\left(a \cdot \frac{c}{b}\right)}^{2} + -1.5 \cdot \left(a \cdot c\right)}}\right)}^{-1}}{3 \cdot a} \]

   4. *-commutative 94.3%

      \[\leadsto \frac{{\left(\frac{b}{\color{blue}{{\left(a \cdot \frac{c}{b}\right)}^{2} \cdot -1.125} + -1.5 \cdot \left(a \cdot c\right)}\right)}^{-1}}{3 \cdot a} \]

   5. fma-define 94.3%

      \[\leadsto \frac{{\left(\frac{b}{\color{blue}{\mathsf{fma}\left({\left(a \cdot \frac{c}{b}\right)}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}}\right)}^{-1}}{3 \cdot a} \]

   6. associate-*r/ 94.3%

      \[\leadsto \frac{{\left(\frac{b}{\mathsf{fma}\left({\color{blue}{\left(\frac{a \cdot c}{b}\right)}}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}\right)}^{-1}}{3 \cdot a} \]

9. Applied egg-rr 94.3%

   \[\leadsto \frac{\color{blue}{{\left(\frac{b}{\mathsf{fma}\left({\left(\frac{a \cdot c}{b}\right)}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}\right)}^{-1}}}{3 \cdot a} \]
10. Step-by-step derivation

    1. unpow-1 94.3%

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left({\left(\frac{a \cdot c}{b}\right)}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]

    2. associate-/l* 94.3%

       \[\leadsto \frac{\frac{1}{\frac{b}{\mathsf{fma}\left({\color{blue}{\left(a \cdot \frac{c}{b}\right)}}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

    3. associate-*r* 94.5%

       \[\leadsto \frac{\frac{1}{\frac{b}{\mathsf{fma}\left({\left(a \cdot \frac{c}{b}\right)}^{2}, -1.125, \color{blue}{\left(-1.5 \cdot a\right) \cdot c}\right)}}}{3 \cdot a} \]

11. Simplified 94.5%

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left({\left(a \cdot \frac{c}{b}\right)}^{2}, -1.125, \left(-1.5 \cdot a\right) \cdot c\right)}}}}{3 \cdot a} \]

12. Taylor expanded in c around 0 94.5%

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]

13. Final simplification 94.5%

    \[\leadsto \frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 3} \]
14. Add Preprocessing

Alternative 10: 94.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}{a}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ 1.0 (/ (+ (* -0.6666666666666666 (/ b c)) (* 0.5 (/ a b))) a))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (1.0 / (((-0.6666666666666666 * (b / c)) + (0.5 * (a / b))) / a)) / (a * 3.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (1.0d0 / ((((-0.6666666666666666d0) * (b / c)) + (0.5d0 * (a / b))) / a)) / (a * 3.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / (((-0.6666666666666666 * (b / c)) + (0.5 * (a / b))) / a)) / (a * 3.0);
}

Python

def code(a, b, c):
	return (1.0 / (((-0.6666666666666666 * (b / c)) + (0.5 * (a / b))) / a)) / (a * 3.0)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(Float64(-0.6666666666666666 * Float64(b / c)) + Float64(0.5 * Float64(a / b))) / a)) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / (((-0.6666666666666666 * (b / c)) + (0.5 * (a / b))) / a)) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[(N[(-0.6666666666666666 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in b around inf 94.4%

   \[\leadsto \frac{\color{blue}{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}}{b}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. associate-/l* 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b}}{3 \cdot a} \]

5. Applied egg-rr 94.4%

   \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}}{b}}{3 \cdot a} \]
6. Step-by-step derivation

   1. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b}}{3 \cdot a} \]

   2. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\left(a \cdot a\right) \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}\right)}{b}}{3 \cdot a} \]

   3. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\left(a \cdot a\right) \cdot \frac{c \cdot c}{\color{blue}{b \cdot b}}\right)}{b}}{3 \cdot a} \]

   4. times-frac 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)}\right)}{b}}{3 \cdot a} \]

   5. swap-sqr 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{\left(\left(a \cdot \frac{c}{b}\right) \cdot \left(a \cdot \frac{c}{b}\right)\right)}}{b}}{3 \cdot a} \]

   6. unpow2 94.4%

      \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{{\left(a \cdot \frac{c}{b}\right)}^{2}}}{b}}{3 \cdot a} \]

7. Simplified 94.4%

   \[\leadsto \frac{\frac{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot \color{blue}{{\left(a \cdot \frac{c}{b}\right)}^{2}}}{b}}{3 \cdot a} \]
8. Step-by-step derivation

   1. clear-num 94.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b}{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot {\left(a \cdot \frac{c}{b}\right)}^{2}}}}}{3 \cdot a} \]

   2. inv-pow 94.3%

      \[\leadsto \frac{\color{blue}{{\left(\frac{b}{-1.5 \cdot \left(a \cdot c\right) + -1.125 \cdot {\left(a \cdot \frac{c}{b}\right)}^{2}}\right)}^{-1}}}{3 \cdot a} \]

   3. +-commutative 94.3%

      \[\leadsto \frac{{\left(\frac{b}{\color{blue}{-1.125 \cdot {\left(a \cdot \frac{c}{b}\right)}^{2} + -1.5 \cdot \left(a \cdot c\right)}}\right)}^{-1}}{3 \cdot a} \]

   4. *-commutative 94.3%

      \[\leadsto \frac{{\left(\frac{b}{\color{blue}{{\left(a \cdot \frac{c}{b}\right)}^{2} \cdot -1.125} + -1.5 \cdot \left(a \cdot c\right)}\right)}^{-1}}{3 \cdot a} \]

   5. fma-define 94.3%

      \[\leadsto \frac{{\left(\frac{b}{\color{blue}{\mathsf{fma}\left({\left(a \cdot \frac{c}{b}\right)}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}}\right)}^{-1}}{3 \cdot a} \]

   6. associate-*r/ 94.3%

      \[\leadsto \frac{{\left(\frac{b}{\mathsf{fma}\left({\color{blue}{\left(\frac{a \cdot c}{b}\right)}}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}\right)}^{-1}}{3 \cdot a} \]

9. Applied egg-rr 94.3%

   \[\leadsto \frac{\color{blue}{{\left(\frac{b}{\mathsf{fma}\left({\left(\frac{a \cdot c}{b}\right)}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}\right)}^{-1}}}{3 \cdot a} \]
10. Step-by-step derivation

    1. unpow-1 94.3%

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left({\left(\frac{a \cdot c}{b}\right)}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]

    2. associate-/l* 94.3%

       \[\leadsto \frac{\frac{1}{\frac{b}{\mathsf{fma}\left({\color{blue}{\left(a \cdot \frac{c}{b}\right)}}^{2}, -1.125, -1.5 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]

    3. associate-*r* 94.5%

       \[\leadsto \frac{\frac{1}{\frac{b}{\mathsf{fma}\left({\left(a \cdot \frac{c}{b}\right)}^{2}, -1.125, \color{blue}{\left(-1.5 \cdot a\right) \cdot c}\right)}}}{3 \cdot a} \]

11. Simplified 94.5%

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left({\left(a \cdot \frac{c}{b}\right)}^{2}, -1.125, \left(-1.5 \cdot a\right) \cdot c\right)}}}}{3 \cdot a} \]

12. Taylor expanded in a around 0 94.5%

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.6666666666666666 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}{a}}}}{3 \cdot a} \]

13. Final simplification 94.5%

    \[\leadsto \frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}{a}}}{a \cdot 3} \]
14. Add Preprocessing

Alternative 11: 90.2% 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 17.1%

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

   1. /-rgt-identity 17.1%

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

   2. metadata-eval 17.1%

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

3. Simplified 17.1%

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

5. Taylor expanded in a around 0 11.9%

   \[\leadsto \frac{\color{blue}{\left(b + -1.5 \cdot \frac{a \cdot c}{b}\right)} - b}{3 \cdot a} \]

6. Taylor expanded in b around 0 90.9%

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

   1. associate-*r/ 90.9%

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

   2. *-commutative 90.9%

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

   3. associate-/l* 90.6%

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

8. Simplified 90.6%

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

9. Final simplification 90.6%

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

Alternative 12: 90.5% 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 17.1%

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

3. Taylor expanded in b around inf 90.9%

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

   1. associate-*r/ 90.9%

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

   2. *-commutative 90.9%

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

5. Simplified 90.9%

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

6. Final simplification 90.9%

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

Reproduce

herbie shell --seed 2024077 
(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)))