Cubic critical, wide range

Percentage Accurate: 18.0% → 99.2%

Time: 15.0s

Alternatives: 7

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: 18.0% 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: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(a \cdot \left(c \cdot 3\right)\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (* a (* c 3.0)) (/ 0.3333333333333333 a))
  (- (- b) (sqrt (fma a (* c -3.0) (pow b 2.0))))))

C

double code(double a, double b, double c) {
	return ((a * (c * 3.0)) * (0.3333333333333333 / a)) / (-b - sqrt(fma(a, (c * -3.0), pow(b, 2.0))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(c * 3.0)) * Float64(0.3333333333333333 / a)) / Float64(Float64(-b) - sqrt(fma(a, Float64(c * -3.0), (b ^ 2.0)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

   1. add-cube-cbrt 17.7%

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

   2. pow3 17.7%

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

   3. associate-*l* 17.7%

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

4. Applied egg-rr 17.7%

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

   1. flip-+ 17.7%

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

   2. pow2 17.7%

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

   3. add-sqr-sqrt 18.2%

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

   4. pow2 18.2%

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

   5. unpow3 18.2%

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

   6. add-cube-cbrt 18.2%

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

   7. *-commutative 18.2%

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

   8. pow2 18.2%

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

   9. unpow3 18.2%

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

   10. add-cube-cbrt 18.2%

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

   11. *-commutative 18.2%

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

6. Applied egg-rr 18.2%

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

   1. associate--r- 99.1%

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

   2. associate-*r* 99.2%

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

   3. associate-*r* 99.2%

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

8. Simplified 99.2%

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

   1. div-inv 99.2%

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

   2. +-commutative 99.2%

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

   3. fma-define 99.2%

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

   4. neg-mul-1 99.2%

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

   5. unpow-prod-down 99.2%

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

   6. metadata-eval 99.2%

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

   7. *-un-lft-identity 99.2%

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

   8. associate-/r* 99.2%

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

   9. metadata-eval 99.2%

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

10. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot 3, {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}} \cdot \frac{0.3333333333333333}{a}} \]
11. Step-by-step derivation

    1. associate-*l/ 99.2%

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

    2. fma-undefine 99.2%

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

    3. +-inverses 99.2%

       \[\leadsto \frac{\left(a \cdot \left(c \cdot 3\right) + \color{blue}{0}\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 3\right)}} \]

    4. +-rgt-identity 99.2%

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

    5. sub-neg 99.2%

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

    6. +-commutative 99.2%

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

    7. distribute-rgt-neg-in 99.2%

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

    8. distribute-rgt-neg-in 99.2%

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

    9. metadata-eval 99.2%

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

    10. *-commutative 99.2%

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

    11. fma-define 99.2%

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

    12. *-commutative 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

    \[\leadsto \frac{\left(a \cdot \left(c \cdot 3\right)\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \]
14. Add Preprocessing

Alternative 2: 95.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in a around 0 95.1%

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

4. Final simplification 95.1%

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

Alternative 3: 95.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (-0.5d0)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 2.0d0)))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in b around inf 95.1%

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

4. Final simplification 95.1%

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

Alternative 4: 95.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in c around 0 94.8%

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

4. Taylor expanded in b around inf 94.8%

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

   1. associate-*r/ 95.1%

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

   2. *-commutative 95.1%

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

   3. fmm-def 95.1%

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

   4. associate-/l* 95.1%

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

   5. metadata-eval 95.1%

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

6. Applied egg-rr 95.1%

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

7. Final simplification 95.1%

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

Alternative 5: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-0.375d0) * ((a * c) / (b ** 3.0d0))) - (0.5d0 / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

   1. add-cube-cbrt 17.7%

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

   2. pow3 17.7%

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

   3. associate-*l* 17.7%

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

4. Applied egg-rr 17.7%

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

   1. flip-+ 17.7%

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

   2. pow2 17.7%

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

   3. add-sqr-sqrt 18.2%

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

   4. pow2 18.2%

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

   5. unpow3 18.2%

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

   6. add-cube-cbrt 18.2%

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

   7. *-commutative 18.2%

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

   8. pow2 18.2%

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

   9. unpow3 18.2%

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

   10. add-cube-cbrt 18.2%

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

   11. *-commutative 18.2%

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

6. Applied egg-rr 18.2%

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

   1. associate--r- 99.1%

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

   2. associate-*r* 99.2%

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

   3. associate-*r* 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in c around 0 94.8%

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

    1. *-commutative 94.8%

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

    2. associate-*r/ 94.8%

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

    3. metadata-eval 94.8%

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

11. Simplified 94.8%

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

12. Final simplification 94.8%

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

Alternative 6: 90.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Simplified 17.7%

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

5. Taylor expanded in a around 0 12.5%

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

   1. div-sub 12.5%

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

   2. +-commutative 12.5%

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

   3. fma-define 12.5%

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

   4. associate-/l* 12.5%

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

7. Applied egg-rr 12.5%

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

   1. div-sub 12.5%

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

   2. *-rgt-identity 12.5%

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

   3. associate-*r/ 12.5%

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

   4. fma-undefine 12.5%

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

   5. associate--l+ 89.9%

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

   6. +-inverses 89.9%

      \[\leadsto \left(-1.5 \cdot \left(a \cdot \frac{c}{b}\right) + \color{blue}{0}\right) \cdot \frac{1}{3 \cdot a} \]

   7. +-rgt-identity 89.9%

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

   8. associate-*r* 90.1%

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

   9. associate-*r/ 89.9%

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

   10. associate-*l/ 89.9%

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

   11. associate-*r/ 89.8%

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

   12. *-commutative 89.8%

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

   13. associate-*l* 89.8%

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

   14. *-commutative 89.8%

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

   15. associate-/r* 89.7%

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

   16. metadata-eval 89.7%

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

9. Simplified 89.7%

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

10. Taylor expanded in a around 0 90.4%

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

    1. associate-*r/ 90.4%

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

    2. *-commutative 90.4%

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

    3. associate-*r/ 90.1%

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

12. Simplified 90.1%

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

13. Final simplification 90.1%

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

Alternative 7: 90.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in b around inf 90.4%

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

   1. associate-*r/ 90.4%

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

   2. *-commutative 90.4%

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

5. Simplified 90.4%

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

6. Final simplification 90.4%

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

Reproduce

herbie shell --seed 2024130 
(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)))