Cubic critical, medium range

Percentage Accurate: 31.2% → 99.7%

Time: 11.5s

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: 31.2% 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: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ c (- (- b) (sqrt (fma -3.0 (* c a) (pow b 2.0))))))

C

double code(double a, double b, double c) {
	return c / (-b - sqrt(fma(-3.0, (c * a), pow(b, 2.0))));
}

Julia

function code(a, b, c)
	return Float64(c / Float64(Float64(-b) - sqrt(fma(-3.0, Float64(c * a), (b ^ 2.0)))))
end

Wolfram

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

Derivation

1. Initial program 34.6%

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

   1. add-cbrt-cube 34.6%

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

   2. pow1/3 34.5%

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

   3. pow3 34.5%

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

   4. associate-*l* 34.6%

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

   5. unpow-prod-down 34.6%

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

   6. metadata-eval 34.6%

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

3. Applied egg-rr 34.6%

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

   1. flip-+ 34.4%

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

   2. pow2 34.4%

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

   3. add-sqr-sqrt 35.3%

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

   4. pow2 35.3%

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

   5. unpow1/3 35.3%

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

   6. *-commutative 35.3%

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

   7. cbrt-prod 35.3%

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

   8. unpow3 35.3%

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

   9. add-cbrt-cube 35.3%

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

5. Applied egg-rr 35.4%

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

   1. associate--r- 98.6%

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

   2. associate-*l* 98.6%

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

   3. associate-*l* 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in a around 0 99.2%

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

   1. *-commutative 99.2%

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

10. Simplified 99.2%

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

12. Applied egg-rr 38.5%

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

    1. expm1-def 85.2%

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

    2. expm1-log1p 99.2%

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

    3. associate-/r* 99.2%

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

    4. fma-udef 99.2%

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

    5. +-inverses 99.2%

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

    6. +-rgt-identity 99.2%

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

    7. times-frac 99.5%

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

    8. metadata-eval 99.5%

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

    9. *-lft-identity 99.5%

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

    10. associate-/l* 99.7%

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

    11. *-inverses 99.7%

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

    12. cancel-sign-sub-inv 99.7%

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

    13. metadata-eval 99.7%

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

    14. *-commutative 99.7%

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

    15. +-commutative 99.7%

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

    16. fma-def 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5e-9)
   (/ (- (sqrt (fma b b (* c (* -3.0 a)))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5e-9) {
		tmp = (sqrt(fma(b, b, (c * (-3.0 * a)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -5e-9)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-3.0 * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -5e-9], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $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)) < -5.0000000000000001e-9

   1. Initial program 70.4%

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

   3. Simplified 70.6%

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

   if -5.0000000000000001e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 16.2%

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

   2. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

      2. associate-*l/ 92.7%

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

   4. Simplified 92.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

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-9) t_0 (/ (* c -0.5) 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-9) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / 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-9)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / 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-9) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / 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-9:
		tmp = t_0
	else:
		tmp = (c * -0.5) / 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-9)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / 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-9)
		tmp = t_0;
	else
		tmp = (c * -0.5) / 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-9], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $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)) < -5.0000000000000001e-9

   1. Initial program 70.4%

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

   if -5.0000000000000001e-9 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 16.2%

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

   2. Taylor expanded in b around inf 92.7%

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

      1. *-commutative 92.7%

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

      2. associate-*l/ 92.7%

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

   4. Simplified 92.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 90.6% 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 34.6%

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

2. Taylor expanded in b around inf 89.2%

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

3. Final simplification 89.2%

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

Alternative 5: 81.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.6%

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

2. Taylor expanded in b around inf 79.0%

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

   1. *-commutative 79.0%

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

   2. associate-*l/ 79.0%

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

4. Simplified 79.0%

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

5. Final simplification 79.0%

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

Reproduce

herbie shell --seed 2023298 
(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)))