Cubic critical, wide range

Percentage Accurate: 17.6% → 99.9%

Time: 9.5s

Alternatives: 5

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: 17.6% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{c}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ c (- (- b) (sqrt (- (pow b 2.0) (* c (* a (cbrt 27.0))))))))

C

double code(double a, double b, double c) {
	return c / (-b - sqrt((pow(b, 2.0) - (c * (a * cbrt(27.0))))));
}

Java

public static double code(double a, double b, double c) {
	return c / (-b - Math.sqrt((Math.pow(b, 2.0) - (c * (a * Math.cbrt(27.0))))));
}

Julia

function code(a, b, c)
	return Float64(c / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - Float64(c * Float64(a * cbrt(27.0)))))))
end

Wolfram

code[a_, b_, c_] := N[(c / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] - N[(c * N[(a * N[Power[27.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.5%

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

   1. add-cbrt-cube 20.5%

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

   2. pow1/3 20.3%

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

   3. pow3 20.3%

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

   4. associate-*l* 20.3%

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

   5. unpow-prod-down 20.3%

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

   6. metadata-eval 20.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

3. Applied egg-rr 20.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. flip-+ 20.3%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}}{3 \cdot a} \]

   2. pow2 20.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}} \cdot \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   3. add-sqr-sqrt 20.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   4. pow2 20.8%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   5. unpow1/3 21.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{27 \cdot {\left(a \cdot c\right)}^{3}}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   6. *-commutative 21.0%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{{\left(a \cdot c\right)}^{3} \cdot 27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   7. cbrt-prod 20.9%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\sqrt[3]{{\left(a \cdot c\right)}^{3}} \cdot \sqrt[3]{27}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   8. unpow3 20.9%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \sqrt[3]{\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right) \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

   9. add-cbrt-cube 20.9%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(a \cdot c\right)} \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]

5. Applied egg-rr 21.0%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}}{3 \cdot a} \]
6. Step-by-step derivation

   1. associate--r- 98.5%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   2. unpow2 98.5%

      \[\leadsto \frac{\frac{\left(\color{blue}{\left(-b\right) \cdot \left(-b\right)} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   3. sqr-neg 98.5%

      \[\leadsto \frac{\frac{\left(\color{blue}{b \cdot b} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   4. unpow2 98.5%

      \[\leadsto \frac{\frac{\left(\color{blue}{{b}^{2}} - {b}^{2}\right) + \left(a \cdot c\right) \cdot \sqrt[3]{27}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   5. associate-*l* 98.5%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \color{blue}{a \cdot \left(c \cdot \sqrt[3]{27}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}}}{3 \cdot a} \]

   6. associate-*l* 98.5%

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]

7. Simplified 98.5%

   \[\leadsto \frac{\color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot \sqrt[3]{27}\right)}}}}{3 \cdot a} \]
8. Step-by-step derivation

   1. expm1-log1p-u 83.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left({b}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot \sqrt[3]{27}\right)}}}{3 \cdot a}\right)\right)} \]

   2. expm1-udef 20.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left({b}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot \sqrt[3]{27}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot \sqrt[3]{27}\right)}}}{3 \cdot a}\right)} - 1} \]

9. Applied egg-rr 20.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\sqrt[3]{27}, a \cdot c, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \sqrt[3]{27} \cdot \left(a \cdot c\right)}\right)}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 83.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\sqrt[3]{27}, a \cdot c, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \sqrt[3]{27} \cdot \left(a \cdot c\right)}\right)}\right)\right)} \]

    2. expm1-log1p 98.6%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{27}, a \cdot c, 0\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \sqrt[3]{27} \cdot \left(a \cdot c\right)}\right)}} \]

    3. associate-/r* 98.6%

       \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt[3]{27}, a \cdot c, 0\right)}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - \sqrt[3]{27} \cdot \left(a \cdot c\right)}}} \]

    4. fma-udef 98.6%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{27} \cdot \left(a \cdot c\right) + 0}}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - \sqrt[3]{27} \cdot \left(a \cdot c\right)}} \]

    5. +-rgt-identity 98.6%

       \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{27} \cdot \left(a \cdot c\right)}}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - \sqrt[3]{27} \cdot \left(a \cdot c\right)}} \]

    6. *-commutative 98.6%

       \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \sqrt[3]{27}}}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - \sqrt[3]{27} \cdot \left(a \cdot c\right)}} \]

    7. *-commutative 98.6%

       \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - \sqrt[3]{27} \cdot \left(a \cdot c\right)}} \]

    8. associate-*l* 98.6%

       \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - \sqrt[3]{27} \cdot \left(a \cdot c\right)}} \]

    9. *-commutative 98.6%

       \[\leadsto \frac{\frac{c \cdot \left(a \cdot \sqrt[3]{27}\right)}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot \sqrt[3]{27}}}} \]

    10. *-commutative 98.6%

        \[\leadsto \frac{\frac{c \cdot \left(a \cdot \sqrt[3]{27}\right)}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(c \cdot a\right)} \cdot \sqrt[3]{27}}} \]

    11. associate-*l* 98.6%

        \[\leadsto \frac{\frac{c \cdot \left(a \cdot \sqrt[3]{27}\right)}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(a \cdot \sqrt[3]{27}\right)}}} \]

11. Simplified 98.6%

    \[\leadsto \color{blue}{\frac{\frac{c \cdot \left(a \cdot \sqrt[3]{27}\right)}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}}} \]

12. Taylor expanded in c around 0 99.8%

    \[\leadsto \frac{\color{blue}{c}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \]

13. Final simplification 99.8%

    \[\leadsto \frac{c}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot \sqrt[3]{27}\right)}} \]

Alternative 2: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -5e-12) t_0 (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -5e-12) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-5d-12)) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -5e-12) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -5e-12:
		tmp = t_0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -5e-12)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -5e-12)
		tmp = t_0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-12], t$95$0, N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4.9999999999999997e-12

   1. Initial program 69.6%

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

   if -4.9999999999999997e-12 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 9.1%

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

   2. Taylor expanded in b around inf 96.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3: 95.5% 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 20.5%

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

2. Taylor expanded in b around inf 93.3%

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

3. Final simplification 93.3%

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

Alternative 4: 90.6% 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 20.5%

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

2. Taylor expanded in b around inf 88.2%

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

3. Final simplification 88.2%

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

Alternative 5: 3.3% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 20.5%

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

   1. add-cbrt-cube 20.5%

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

   2. pow1/3 20.3%

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

   3. pow3 20.3%

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

   4. associate-*l* 20.3%

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

   5. unpow-prod-down 20.3%

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

   6. metadata-eval 20.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\color{blue}{27} \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

3. Applied egg-rr 20.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
4. Step-by-step derivation

   1. expm1-log1p-u 13.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a}\right)\right)} \]

   2. expm1-udef 8.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - {\left(27 \cdot {\left(a \cdot c\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a}\right)} - 1} \]

5. Applied egg-rr 8.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right)}{a \cdot 3}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 13.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right)}{a \cdot 3}\right)\right)} \]

   2. expm1-log1p 20.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right)}{a \cdot 3}} \]

   3. *-lft-identity 20.4%

      \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right)}}{a \cdot 3} \]

   4. associate-*l/ 20.4%

      \[\leadsto \color{blue}{\frac{1}{a \cdot 3} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right)} \]

   5. *-commutative 20.4%

      \[\leadsto \frac{1}{\color{blue}{3 \cdot a}} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right) \]

   6. associate-/r* 20.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{a}} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right) \]

   7. metadata-eval 20.4%

      \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right) \]

   8. fma-udef 20.4%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1 \cdot b + \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right)} \]

   9. *-commutative 20.4%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{b \cdot -1} + \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right) \]

   10. fma-def 20.4%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\mathsf{fma}\left(b, -1, \sqrt{{b}^{2} - \left(a \cdot c\right) \cdot \sqrt[3]{27}}\right)} \]

   11. associate-*l* 20.4%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(b, -1, \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot \sqrt[3]{27}\right)}}\right) \]

7. Simplified 20.4%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(b, -1, \sqrt{{b}^{2} - a \cdot \left(c \cdot \sqrt[3]{27}\right)}\right)} \]

8. Taylor expanded in a around 0 3.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 3.3%

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

   2. distribute-rgt1-in 3.3%

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

   3. metadata-eval 3.3%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.3%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.3%

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

10. Simplified 3.3%

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

11. Final simplification 3.3%

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

Reproduce

herbie shell --seed 2023321 
(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)))