Cubic critical, medium range

Percentage Accurate: 31.6% → 99.8%

Time: 11.8s

Alternatives: 5

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.6%

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

   1. neg-sub0 29.6%

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

   2. associate-+l- 29.6%

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

   3. sub0-neg 29.6%

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

   4. neg-mul-1 29.6%

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

   5. associate-*r/ 29.6%

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

   6. metadata-eval 29.6%

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

   7. metadata-eval 29.6%

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

   8. times-frac 29.6%

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

   9. *-commutative 29.6%

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

   10. times-frac 29.6%

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

   11. associate-*l/ 29.6%

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

3. Simplified 29.6%

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

   1. add-log-exp 10.9%

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

   2. associate-*r* 10.9%

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

   3. exp-prod 8.2%

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

   4. *-commutative 8.2%

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

   5. exp-prod 8.1%

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

5. Applied egg-rr 8.1%

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

   1. log-pow 14.9%

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

   2. log-pow 15.5%

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

7. Simplified 15.5%

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

   1. flip-+ 15.5%

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

   2. add-sqr-sqrt 16.0%

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

   3. add-log-exp 17.5%

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

   4. *-commutative 17.5%

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

   5. add-log-exp 30.5%

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

   6. *-commutative 30.5%

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

9. Applied egg-rr 30.5%

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

    1. *-un-lft-identity 30.5%

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

    2. associate-/l/ 30.5%

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

    3. sqr-neg 30.5%

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

    4. associate--r- 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. div-inv 99.2%

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

    2. +-inverses 99.2%

       \[\leadsto 1 \cdot \left(\left(\color{blue}{0} + c \cdot \left(a \cdot 3\right)\right) \cdot \frac{1}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}\right) \]

13. Applied egg-rr 99.2%

    \[\leadsto 1 \cdot \color{blue}{\left(\left(0 + c \cdot \left(a \cdot 3\right)\right) \cdot \frac{1}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}\right)} \]
14. Step-by-step derivation

    1. +-lft-identity 99.2%

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

    2. associate-*r/ 99.3%

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

    3. *-rgt-identity 99.3%

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

    4. associate-/r* 99.6%

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

    5. associate-*r* 99.3%

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

    6. times-frac 99.6%

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

    7. associate-/l* 99.7%

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

    8. *-inverses 99.7%

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

    9. metadata-eval 99.7%

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

    10. fma-neg 99.8%

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

    11. associate-*r* 99.8%

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

    12. distribute-rgt-neg-in 99.8%

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

    13. metadata-eval 99.8%

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

15. Simplified 99.8%

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

16. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ t_0 \cdot \frac{\frac{1}{a \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - t_0}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))))
   (* t_0 (/ (/ 1.0 (* a 3.0)) (- (- b) (sqrt (- (* b b) t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return t_0 * ((1.0 / (a * 3.0)) / (-b - sqrt(((b * b) - t_0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = c * (a * 3.0d0)
    code = t_0 * ((1.0d0 / (a * 3.0d0)) / (-b - sqrt(((b * b) - t_0))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return t_0 * ((1.0 / (a * 3.0)) / (-b - Math.sqrt(((b * b) - t_0))));
}

Python

def code(a, b, c):
	t_0 = c * (a * 3.0)
	return t_0 * ((1.0 / (a * 3.0)) / (-b - math.sqrt(((b * b) - t_0))))

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 3.0))
	return Float64(t_0 * Float64(Float64(1.0 / Float64(a * 3.0)) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_0)))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (a * 3.0);
	tmp = t_0 * ((1.0 / (a * 3.0)) / (-b - sqrt(((b * b) - t_0))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(N[(1.0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.6%

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

   1. neg-sub0 29.6%

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

   2. associate-+l- 29.6%

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

   3. sub0-neg 29.6%

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

   4. neg-mul-1 29.6%

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

   5. associate-*r/ 29.6%

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

   6. metadata-eval 29.6%

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

   7. metadata-eval 29.6%

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

   8. times-frac 29.6%

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

   9. *-commutative 29.6%

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

   10. times-frac 29.6%

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

   11. associate-*l/ 29.6%

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

3. Simplified 29.6%

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

   1. add-log-exp 10.9%

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

   2. associate-*r* 10.9%

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

   3. exp-prod 8.2%

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

   4. *-commutative 8.2%

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

   5. exp-prod 8.1%

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

5. Applied egg-rr 8.1%

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

   1. log-pow 14.9%

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

   2. log-pow 15.5%

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

7. Simplified 15.5%

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

   1. flip-+ 15.5%

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

   2. add-sqr-sqrt 16.0%

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

   3. add-log-exp 17.5%

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

   4. *-commutative 17.5%

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

   5. add-log-exp 30.5%

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

   6. *-commutative 30.5%

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

9. Applied egg-rr 30.5%

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

    1. *-un-lft-identity 30.5%

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

    2. associate-/l/ 30.5%

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

    3. sqr-neg 30.5%

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

    4. associate--r- 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. div-inv 99.2%

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

    2. +-inverses 99.2%

       \[\leadsto 1 \cdot \left(\left(\color{blue}{0} + c \cdot \left(a \cdot 3\right)\right) \cdot \frac{1}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}\right) \]

13. Applied egg-rr 99.2%

    \[\leadsto 1 \cdot \color{blue}{\left(\left(0 + c \cdot \left(a \cdot 3\right)\right) \cdot \frac{1}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}\right)} \]
14. Step-by-step derivation

    1. +-lft-identity 99.2%

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

    2. associate-/r* 99.3%

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

15. Simplified 99.3%

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

16. Final simplification 99.3%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ \frac{t_0}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - t_0}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))))
   (/ t_0 (* (* a 3.0) (- (- b) (sqrt (- (* b b) t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return t_0 / ((a * 3.0) * (-b - sqrt(((b * b) - t_0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = c * (a * 3.0d0)
    code = t_0 / ((a * 3.0d0) * (-b - sqrt(((b * b) - t_0))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return t_0 / ((a * 3.0) * (-b - Math.sqrt(((b * b) - t_0))));
}

Python

def code(a, b, c):
	t_0 = c * (a * 3.0)
	return t_0 / ((a * 3.0) * (-b - math.sqrt(((b * b) - t_0))))

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 3.0))
	return Float64(t_0 / Float64(Float64(a * 3.0) * Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - t_0)))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (a * 3.0);
	tmp = t_0 / ((a * 3.0) * (-b - sqrt(((b * b) - t_0))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 / N[(N[(a * 3.0), $MachinePrecision] * N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.6%

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

   1. neg-sub0 29.6%

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

   2. associate-+l- 29.6%

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

   3. sub0-neg 29.6%

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

   4. neg-mul-1 29.6%

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

   5. associate-*r/ 29.6%

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

   6. metadata-eval 29.6%

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

   7. metadata-eval 29.6%

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

   8. times-frac 29.6%

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

   9. *-commutative 29.6%

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

   10. times-frac 29.6%

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

   11. associate-*l/ 29.6%

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

3. Simplified 29.6%

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

   1. add-log-exp 10.9%

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

   2. associate-*r* 10.9%

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

   3. exp-prod 8.2%

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

   4. *-commutative 8.2%

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

   5. exp-prod 8.1%

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

5. Applied egg-rr 8.1%

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

   1. log-pow 14.9%

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

   2. log-pow 15.5%

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

7. Simplified 15.5%

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

   1. flip-+ 15.5%

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

   2. add-sqr-sqrt 16.0%

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

   3. add-log-exp 17.5%

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

   4. *-commutative 17.5%

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

   5. add-log-exp 30.5%

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

   6. *-commutative 30.5%

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

9. Applied egg-rr 30.5%

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

    1. *-un-lft-identity 30.5%

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

    2. associate-/l/ 30.5%

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

    3. sqr-neg 30.5%

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

    4. associate--r- 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. +-inverses 99.3%

       \[\leadsto 1 \cdot \left(1 \cdot \frac{\color{blue}{0} + c \cdot \left(a \cdot 3\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}\right) \]

13. Applied egg-rr 99.3%

    \[\leadsto 1 \cdot \color{blue}{\left(1 \cdot \frac{0 + c \cdot \left(a \cdot 3\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}\right)} \]

14. Final simplification 99.3%

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

Alternative 4: 90.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(a \cdot 3\right) + \left(b \cdot b - b \cdot b\right)}{\left(a \cdot 3\right) \cdot \left(1.5 \cdot \frac{c \cdot a}{b} + b \cdot -2\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+ (* c (* a 3.0)) (- (* b b) (* b b)))
  (* (* a 3.0) (+ (* 1.5 (/ (* c a) b)) (* b -2.0)))))

C

double code(double a, double b, double c) {
	return ((c * (a * 3.0)) + ((b * b) - (b * b))) / ((a * 3.0) * ((1.5 * ((c * a) / b)) + (b * -2.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (a * 3.0d0)) + ((b * b) - (b * b))) / ((a * 3.0d0) * ((1.5d0 * ((c * a) / b)) + (b * (-2.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * (a * 3.0)) + ((b * b) - (b * b))) / ((a * 3.0) * ((1.5 * ((c * a) / b)) + (b * -2.0)));
}

Python

def code(a, b, c):
	return ((c * (a * 3.0)) + ((b * b) - (b * b))) / ((a * 3.0) * ((1.5 * ((c * a) / b)) + (b * -2.0)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(a * 3.0)) + Float64(Float64(b * b) - Float64(b * b))) / Float64(Float64(a * 3.0) * Float64(Float64(1.5 * Float64(Float64(c * a) / b)) + Float64(b * -2.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * (a * 3.0)) + ((b * b) - (b * b))) / ((a * 3.0) * ((1.5 * ((c * a) / b)) + (b * -2.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * 3.0), $MachinePrecision] * N[(N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.6%

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

   1. neg-sub0 29.6%

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

   2. associate-+l- 29.6%

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

   3. sub0-neg 29.6%

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

   4. neg-mul-1 29.6%

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

   5. associate-*r/ 29.6%

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

   6. metadata-eval 29.6%

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

   7. metadata-eval 29.6%

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

   8. times-frac 29.6%

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

   9. *-commutative 29.6%

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

   10. times-frac 29.6%

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

   11. associate-*l/ 29.6%

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

3. Simplified 29.6%

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

   1. add-log-exp 10.9%

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

   2. associate-*r* 10.9%

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

   3. exp-prod 8.2%

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

   4. *-commutative 8.2%

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

   5. exp-prod 8.1%

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

5. Applied egg-rr 8.1%

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

   1. log-pow 14.9%

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

   2. log-pow 15.5%

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

7. Simplified 15.5%

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

   1. flip-+ 15.5%

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

   2. add-sqr-sqrt 16.0%

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

   3. add-log-exp 17.5%

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

   4. *-commutative 17.5%

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

   5. add-log-exp 30.5%

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

   6. *-commutative 30.5%

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

9. Applied egg-rr 30.5%

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

    1. *-un-lft-identity 30.5%

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

    2. associate-/l/ 30.5%

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

    3. sqr-neg 30.5%

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

    4. associate--r- 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

12. Taylor expanded in b around inf 91.3%

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

13. Final simplification 91.3%

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

Alternative 5: 81.1% 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 29.6%

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

   1. /-rgt-identity 29.6%

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

   2. metadata-eval 29.6%

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

   3. associate-/r/ 29.6%

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

   4. metadata-eval 29.6%

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

   5. metadata-eval 29.6%

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

   6. times-frac 29.6%

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

   7. *-commutative 29.6%

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

   8. times-frac 29.6%

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

   9. associate-/r* 29.6%

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

3. Simplified 29.6%

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

4. Taylor expanded in b around inf 82.7%

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

5. Final simplification 82.7%

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

Reproduce

herbie shell --seed 2023174 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))