Cubic critical, wide range

Percentage Accurate: 17.9% → 99.4%

Time: 8.1s

Alternatives: 3

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: 17.9% 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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.6%

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

   1. neg-sub0 16.6%

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

   2. sqr-neg 16.6%

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

   3. associate-+l- 16.6%

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

   4. sub0-neg 16.6%

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

3. Simplified 16.6%

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

   1. add-cbrt-cube 16.7%

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

   2. pow3 16.8%

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

   3. pow1/3 18.6%

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

   4. sqrt-pow2 18.6%

      \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. metadata-eval 18.6%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

5. Applied egg-rr 18.6%

   \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
6. Step-by-step derivation

   1. unpow1/3 16.6%

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

   2. fma-udef 16.6%

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

   3. associate-*r* 16.6%

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

   4. *-commutative 16.6%

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

   5. +-commutative 16.6%

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

   6. *-commutative 16.6%

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

   7. associate-*r* 16.6%

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

   8. fma-def 16.6%

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

7. Simplified 16.6%

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

   1. flip-- 16.6%

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

9. Applied egg-rr 16.6%

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

11. Simplified 17.0%

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

12. Taylor expanded in c around 0 99.1%

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

    1. *-commutative 99.1%

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

    2. associate-*r* 99.3%

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

14. Simplified 99.3%

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

15. Final simplification 99.3%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	return (t_0 / (b + sqrt((t_0 + (b * b))))) / (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
    real(8) :: t_0
    t_0 = c * (a * (-3.0d0))
    code = (t_0 / (b + sqrt((t_0 + (b * b))))) / (a * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.6%

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

   1. neg-sub0 16.6%

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

   2. sqr-neg 16.6%

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

   3. associate-+l- 16.6%

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

   4. sub0-neg 16.6%

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

3. Simplified 16.6%

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

   1. add-cbrt-cube 16.7%

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

   2. pow3 16.8%

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

   3. pow1/3 18.6%

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

   4. sqrt-pow2 18.6%

      \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b}{3 \cdot a} \]

   5. metadata-eval 18.6%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b}{3 \cdot a} \]

5. Applied egg-rr 18.6%

   \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b}{3 \cdot a} \]
6. Step-by-step derivation

   1. unpow1/3 16.6%

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

   2. fma-udef 16.6%

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

   3. associate-*r* 16.6%

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

   4. *-commutative 16.6%

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

   5. +-commutative 16.6%

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

   6. *-commutative 16.6%

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

   7. associate-*r* 16.6%

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

   8. fma-def 16.6%

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

7. Simplified 16.6%

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

   1. flip-- 16.6%

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

9. Applied egg-rr 16.6%

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

11. Simplified 17.0%

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

12. Taylor expanded in c around 0 99.1%

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

    1. *-commutative 99.1%

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

    2. associate-*r* 99.3%

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

14. Simplified 99.3%

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

    1. fma-udef 99.3%

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

    2. associate-*r* 99.3%

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

16. Applied egg-rr 99.3%

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

17. Final simplification 99.3%

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

Alternative 3: 90.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.6%

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

2. Taylor expanded in b around inf 91.2%

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

3. Final simplification 91.2%

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

Reproduce

herbie shell --seed 2023272 
(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)))