Quadratic roots, wide range

Percentage Accurate: 17.7% → 99.9%

Time: 14.5s

Alternatives: 7

Speedup: 29.0×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 17.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot 2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* c 2.0) (- (- b) (sqrt (fma -4.0 (* c a) (pow b 2.0))))))

C

double code(double a, double b, double c) {
	return (c * 2.0) / (-b - sqrt(fma(-4.0, (c * a), pow(b, 2.0))));
}

Julia

function code(a, b, c)
	return Float64(Float64(c * 2.0) / Float64(Float64(-b) - sqrt(fma(-4.0, Float64(c * a), (b ^ 2.0)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow3 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c}\right)}^{3}}}}{a \cdot 2} \]

   3. associate-*l* 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}^{3}}}{a \cdot 2} \]

6. Applied egg-rr 18.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]
7. Step-by-step derivation

   1. flip-+ 18.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}}{a \cdot 2} \]

   2. pow2 18.7%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   3. add-sqr-sqrt 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   4. pow2 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   5. unpow3 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   6. add-cube-cbrt 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. associate-*r* 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(4 \cdot a\right) \cdot c}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   8. pow2 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   9. unpow3 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}}{a \cdot 2} \]

   10. add-cube-cbrt 19.2%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   11. associate-*r* 19.2%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

8. Applied egg-rr 19.2%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. associate--r- 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   2. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. *-commutative 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{\left(a \cdot c\right) \cdot 4}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{a \cdot \left(c \cdot 4\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   6. *-commutative 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}}{a \cdot 2} \]

   7. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]

10. Simplified 99.4%

    \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. div-inv 99.3%

       \[\leadsto \color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2}} \]

    2. +-commutative 99.3%

       \[\leadsto \frac{\color{blue}{a \cdot \left(c \cdot 4\right) + \left({\left(-b\right)}^{2} - {b}^{2}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    3. fma-define 99.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot 4, {\left(-b\right)}^{2} - {b}^{2}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    4. neg-mul-1 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, {\color{blue}{\left(-1 \cdot b\right)}}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    5. unpow-prod-down 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{-1}^{2} \cdot {b}^{2}} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    6. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{1} \cdot {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    7. *-un-lft-identity 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{b}^{2}} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    8. *-commutative 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

12. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{2 \cdot a}} \]
13. Step-by-step derivation

    1. associate-*l/ 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}} \]

    2. fma-undefine 99.5%

       \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    3. +-inverses 99.5%

       \[\leadsto \frac{\left(a \cdot \left(c \cdot 4\right) + \color{blue}{0}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    4. +-rgt-identity 99.5%

       \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot 4\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    5. associate-*r* 99.5%

       \[\leadsto \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot 4\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    6. *-commutative 99.5%

       \[\leadsto \frac{\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    7. associate-/r* 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    8. metadata-eval 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{\color{blue}{0.5}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    9. associate-*r* 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}} \]

    10. *-commutative 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}} \]

    11. sub-neg 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-4 \cdot \left(a \cdot c\right)\right)}}} \]

    12. +-commutative 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + {b}^{2}}}} \]

    13. distribute-lft-neg-in 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)} + {b}^{2}}} \]

    14. metadata-eval 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{-4} \cdot \left(a \cdot c\right) + {b}^{2}}} \]

    15. fma-define 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}} \]

14. Simplified 99.5%

    \[\leadsto \color{blue}{\frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}} \]

15. Taylor expanded in a around 0 99.9%

    \[\leadsto \frac{\color{blue}{2 \cdot c}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}} \]
16. Step-by-step derivation

    1. *-commutative 99.9%

       \[\leadsto \frac{\color{blue}{c \cdot 2}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}} \]

17. Simplified 99.9%

    \[\leadsto \frac{\color{blue}{c \cdot 2}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}} \]

18. Final simplification 99.9%

    \[\leadsto \frac{c \cdot 2}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}} \]
19. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(c \cdot a\right) \cdot 4\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (* (* c a) 4.0) (/ 0.5 a))
  (- (- b) (sqrt (* a (+ (* c -4.0) (/ (pow b 2.0) a)))))))

C

double code(double a, double b, double c) {
	return (((c * a) * 4.0) * (0.5 / a)) / (-b - sqrt((a * ((c * -4.0) + (pow(b, 2.0) / 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 = (((c * a) * 4.0d0) * (0.5d0 / a)) / (-b - sqrt((a * ((c * (-4.0d0)) + ((b ** 2.0d0) / a)))))
end function

Java

public static double code(double a, double b, double c) {
	return (((c * a) * 4.0) * (0.5 / a)) / (-b - Math.sqrt((a * ((c * -4.0) + (Math.pow(b, 2.0) / a)))));
}

Python

def code(a, b, c):
	return (((c * a) * 4.0) * (0.5 / a)) / (-b - math.sqrt((a * ((c * -4.0) + (math.pow(b, 2.0) / a)))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(c * a) * 4.0) * Float64(0.5 / a)) / Float64(Float64(-b) - sqrt(Float64(a * Float64(Float64(c * -4.0) + Float64((b ^ 2.0) / a))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * a) * 4.0) * (0.5 / a)) / (-b - sqrt((a * ((c * -4.0) + ((b ^ 2.0) / a)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(a * N[(N[(c * -4.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow3 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c}\right)}^{3}}}}{a \cdot 2} \]

   3. associate-*l* 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}^{3}}}{a \cdot 2} \]

6. Applied egg-rr 18.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]
7. Step-by-step derivation

   1. flip-+ 18.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}}{a \cdot 2} \]

   2. pow2 18.7%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   3. add-sqr-sqrt 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   4. pow2 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   5. unpow3 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   6. add-cube-cbrt 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. associate-*r* 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(4 \cdot a\right) \cdot c}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   8. pow2 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   9. unpow3 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}}{a \cdot 2} \]

   10. add-cube-cbrt 19.2%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   11. associate-*r* 19.2%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

8. Applied egg-rr 19.2%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. associate--r- 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   2. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. *-commutative 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{\left(a \cdot c\right) \cdot 4}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{a \cdot \left(c \cdot 4\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   6. *-commutative 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}}{a \cdot 2} \]

   7. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]

10. Simplified 99.4%

    \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. div-inv 99.3%

       \[\leadsto \color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2}} \]

    2. +-commutative 99.3%

       \[\leadsto \frac{\color{blue}{a \cdot \left(c \cdot 4\right) + \left({\left(-b\right)}^{2} - {b}^{2}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    3. fma-define 99.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot 4, {\left(-b\right)}^{2} - {b}^{2}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    4. neg-mul-1 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, {\color{blue}{\left(-1 \cdot b\right)}}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    5. unpow-prod-down 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{-1}^{2} \cdot {b}^{2}} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    6. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{1} \cdot {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    7. *-un-lft-identity 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{b}^{2}} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    8. *-commutative 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

12. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{2 \cdot a}} \]
13. Step-by-step derivation

    1. associate-*l/ 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}} \]

    2. fma-undefine 99.5%

       \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    3. +-inverses 99.5%

       \[\leadsto \frac{\left(a \cdot \left(c \cdot 4\right) + \color{blue}{0}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    4. +-rgt-identity 99.5%

       \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot 4\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    5. associate-*r* 99.5%

       \[\leadsto \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot 4\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    6. *-commutative 99.5%

       \[\leadsto \frac{\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    7. associate-/r* 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    8. metadata-eval 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{\color{blue}{0.5}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    9. associate-*r* 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}} \]

    10. *-commutative 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}} \]

    11. sub-neg 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-4 \cdot \left(a \cdot c\right)\right)}}} \]

    12. +-commutative 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + {b}^{2}}}} \]

    13. distribute-lft-neg-in 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)} + {b}^{2}}} \]

    14. metadata-eval 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{-4} \cdot \left(a \cdot c\right) + {b}^{2}}} \]

    15. fma-define 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}} \]

14. Simplified 99.5%

    \[\leadsto \color{blue}{\frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}} \]

15. Taylor expanded in a around inf 99.5%

    \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{a \cdot \left(-4 \cdot c + \frac{{b}^{2}}{a}\right)}}} \]

16. Final simplification 99.5%

    \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 4\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{a \cdot \left(c \cdot -4 + \frac{{b}^{2}}{a}\right)}} \]
17. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(c \cdot a\right) \cdot 4\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} + -4 \cdot \left(c \cdot a\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (* (* c a) 4.0) (/ 0.5 a))
  (- (- b) (sqrt (+ (pow b 2.0) (* -4.0 (* c a)))))))

C

double code(double a, double b, double c) {
	return (((c * a) * 4.0) * (0.5 / a)) / (-b - sqrt((pow(b, 2.0) + (-4.0 * (c * 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 = (((c * a) * 4.0d0) * (0.5d0 / a)) / (-b - sqrt(((b ** 2.0d0) + ((-4.0d0) * (c * a)))))
end function

Java

public static double code(double a, double b, double c) {
	return (((c * a) * 4.0) * (0.5 / a)) / (-b - Math.sqrt((Math.pow(b, 2.0) + (-4.0 * (c * a)))));
}

Python

def code(a, b, c):
	return (((c * a) * 4.0) * (0.5 / a)) / (-b - math.sqrt((math.pow(b, 2.0) + (-4.0 * (c * a)))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(c * a) * 4.0) * Float64(0.5 / a)) / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) + Float64(-4.0 * Float64(c * a))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * a) * 4.0) * (0.5 / a)) / (-b - sqrt(((b ^ 2.0) + (-4.0 * (c * a)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[Power[b, 2.0], $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow3 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c}\right)}^{3}}}}{a \cdot 2} \]

   3. associate-*l* 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}^{3}}}{a \cdot 2} \]

6. Applied egg-rr 18.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]
7. Step-by-step derivation

   1. flip-+ 18.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}}{a \cdot 2} \]

   2. pow2 18.7%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   3. add-sqr-sqrt 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   4. pow2 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   5. unpow3 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   6. add-cube-cbrt 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. associate-*r* 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(4 \cdot a\right) \cdot c}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   8. pow2 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   9. unpow3 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}}{a \cdot 2} \]

   10. add-cube-cbrt 19.2%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   11. associate-*r* 19.2%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

8. Applied egg-rr 19.2%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. associate--r- 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   2. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. *-commutative 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{\left(a \cdot c\right) \cdot 4}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{a \cdot \left(c \cdot 4\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   6. *-commutative 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}}{a \cdot 2} \]

   7. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]

10. Simplified 99.4%

    \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. div-inv 99.3%

       \[\leadsto \color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2}} \]

    2. +-commutative 99.3%

       \[\leadsto \frac{\color{blue}{a \cdot \left(c \cdot 4\right) + \left({\left(-b\right)}^{2} - {b}^{2}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    3. fma-define 99.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot 4, {\left(-b\right)}^{2} - {b}^{2}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    4. neg-mul-1 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, {\color{blue}{\left(-1 \cdot b\right)}}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    5. unpow-prod-down 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{-1}^{2} \cdot {b}^{2}} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    6. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{1} \cdot {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    7. *-un-lft-identity 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{b}^{2}} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    8. *-commutative 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

12. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{2 \cdot a}} \]
13. Step-by-step derivation

    1. associate-*l/ 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}} \]

    2. fma-undefine 99.5%

       \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    3. +-inverses 99.5%

       \[\leadsto \frac{\left(a \cdot \left(c \cdot 4\right) + \color{blue}{0}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    4. +-rgt-identity 99.5%

       \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot 4\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    5. associate-*r* 99.5%

       \[\leadsto \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot 4\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    6. *-commutative 99.5%

       \[\leadsto \frac{\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    7. associate-/r* 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    8. metadata-eval 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{\color{blue}{0.5}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    9. associate-*r* 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}} \]

    10. *-commutative 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}} \]

    11. sub-neg 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-4 \cdot \left(a \cdot c\right)\right)}}} \]

    12. +-commutative 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + {b}^{2}}}} \]

    13. distribute-lft-neg-in 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)} + {b}^{2}}} \]

    14. metadata-eval 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{-4} \cdot \left(a \cdot c\right) + {b}^{2}}} \]

    15. fma-define 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}} \]

14. Simplified 99.5%

    \[\leadsto \color{blue}{\frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}} \]

15. Taylor expanded in a around 0 99.5%

    \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}} \]

16. Final simplification 99.5%

    \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 4\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} + -4 \cdot \left(c \cdot a\right)}} \]
17. Add Preprocessing

Alternative 4: 95.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c / -b) - (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 = (c / -b) - (a * ((c ** 2.0d0) / (b ** 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 95.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
6. Step-by-step derivation

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

   3. mul-1-neg 95.7%

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

   4. distribute-neg-frac2 95.7%

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

   5. associate-/l* 95.7%

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

7. Simplified 95.7%

   \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]
8. Add Preprocessing

Alternative 5: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((pow((c / -b), 2.0) * -a) - 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 = ((((c / -b) ** 2.0d0) * -a) - c) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 97.4%

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]

6. Taylor expanded in a around 0 95.7%

   \[\leadsto \frac{\color{blue}{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}}{b} \]
7. Step-by-step derivation

   1. neg-mul-1 95.7%

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

   2. +-commutative 95.7%

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

   3. unsub-neg 95.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} - c}}{b} \]

   4. mul-1-neg 95.7%

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

   5. associate-*r/ 95.7%

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

   6. *-commutative 95.7%

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

   7. distribute-rgt-neg-in 95.7%

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

   8. unpow2 95.7%

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

   9. unpow2 95.7%

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

   10. times-frac 95.7%

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

   11. sqr-neg 95.7%

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

   12. distribute-frac-neg 95.7%

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

   13. distribute-frac-neg 95.7%

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

   14. unpow2 95.7%

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

8. Simplified 95.7%

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

9. Final simplification 95.7%

   \[\leadsto \frac{{\left(\frac{c}{-b}\right)}^{2} \cdot \left(-a\right) - c}{b} \]
10. Add Preprocessing

Alternative 6: 95.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(c \cdot a\right) \cdot 4\right) \cdot \frac{0.5}{a}}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (* (* c a) 4.0) (/ 0.5 a)) (* 2.0 (- (* a (/ c b)) b))))

C

double code(double a, double b, double c) {
	return (((c * a) * 4.0) * (0.5 / a)) / (2.0 * ((a * (c / b)) - 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 * a) * 4.0d0) * (0.5d0 / a)) / (2.0d0 * ((a * (c / b)) - b))
end function

Java

public static double code(double a, double b, double c) {
	return (((c * a) * 4.0) * (0.5 / a)) / (2.0 * ((a * (c / b)) - b));
}

Python

def code(a, b, c):
	return (((c * a) * 4.0) * (0.5 / a)) / (2.0 * ((a * (c / b)) - b))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(c * a) * 4.0) * Float64(0.5 / a)) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * a) * 4.0) * (0.5 / a)) / (2.0 * ((a * (c / b)) - b));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(c * a), $MachinePrecision] * 4.0), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow3 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{\left(4 \cdot a\right) \cdot c}\right)}^{3}}}}{a \cdot 2} \]

   3. associate-*l* 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{4 \cdot \left(a \cdot c\right)}}\right)}^{3}}}{a \cdot 2} \]

6. Applied egg-rr 18.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]
7. Step-by-step derivation

   1. flip-+ 18.7%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}}{a \cdot 2} \]

   2. pow2 18.7%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}} \cdot \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   3. add-sqr-sqrt 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}\right)}}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   4. pow2 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   5. unpow3 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   6. add-cube-cbrt 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. associate-*r* 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{\left(4 \cdot a\right) \cdot c}\right)}{\left(-b\right) - \sqrt{b \cdot b - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   8. pow2 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - {\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

   9. unpow3 19.2%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{4 \cdot \left(a \cdot c\right)}}}}}{a \cdot 2} \]

   10. add-cube-cbrt 19.2%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   11. associate-*r* 19.2%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

8. Applied egg-rr 19.2%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]
9. Step-by-step derivation

   1. associate--r- 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   2. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. *-commutative 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{\left(a \cdot c\right) \cdot 4}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + \color{blue}{a \cdot \left(c \cdot 4\right)}}{\left(-b\right) - \sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

   6. *-commutative 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}}}{a \cdot 2} \]

   7. associate-*r* 99.4%

      \[\leadsto \frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]

10. Simplified 99.4%

    \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. div-inv 99.3%

       \[\leadsto \color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(c \cdot 4\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2}} \]

    2. +-commutative 99.3%

       \[\leadsto \frac{\color{blue}{a \cdot \left(c \cdot 4\right) + \left({\left(-b\right)}^{2} - {b}^{2}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    3. fma-define 99.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot 4, {\left(-b\right)}^{2} - {b}^{2}\right)}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    4. neg-mul-1 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, {\color{blue}{\left(-1 \cdot b\right)}}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    5. unpow-prod-down 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{-1}^{2} \cdot {b}^{2}} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    6. metadata-eval 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{1} \cdot {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    7. *-un-lft-identity 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, \color{blue}{{b}^{2}} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{a \cdot 2} \]

    8. *-commutative 99.3%

       \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

12. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \cdot \frac{1}{2 \cdot a}} \]
13. Step-by-step derivation

    1. associate-*l/ 99.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot 4, {b}^{2} - {b}^{2}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}}} \]

    2. fma-undefine 99.5%

       \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot 4\right) + \left({b}^{2} - {b}^{2}\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    3. +-inverses 99.5%

       \[\leadsto \frac{\left(a \cdot \left(c \cdot 4\right) + \color{blue}{0}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    4. +-rgt-identity 99.5%

       \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot 4\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    5. associate-*r* 99.5%

       \[\leadsto \frac{\color{blue}{\left(\left(a \cdot c\right) \cdot 4\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    6. *-commutative 99.5%

       \[\leadsto \frac{\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    7. associate-/r* 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    8. metadata-eval 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{\color{blue}{0.5}}{a}}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(c \cdot 4\right)}} \]

    9. associate-*r* 99.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{\left(a \cdot c\right) \cdot 4}}} \]

    10. *-commutative 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}} \]

    11. sub-neg 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-4 \cdot \left(a \cdot c\right)\right)}}} \]

    12. +-commutative 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\left(-4 \cdot \left(a \cdot c\right)\right) + {b}^{2}}}} \]

    13. distribute-lft-neg-in 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)} + {b}^{2}}} \]

    14. metadata-eval 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{-4} \cdot \left(a \cdot c\right) + {b}^{2}}} \]

    15. fma-define 99.5%

        \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}} \]

14. Simplified 99.5%

    \[\leadsto \color{blue}{\frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, {b}^{2}\right)}}} \]

15. Taylor expanded in a around 0 95.5%

    \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}} \]
16. Step-by-step derivation

    1. distribute-lft-out-- 95.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}} \]

    2. associate-/l* 95.5%

       \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} - b\right)} \]

17. Simplified 95.5%

    \[\leadsto \frac{\left(4 \cdot \left(a \cdot c\right)\right) \cdot \frac{0.5}{a}}{\color{blue}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}} \]

18. Final simplification 95.5%

    \[\leadsto \frac{\left(\left(c \cdot a\right) \cdot 4\right) \cdot \frac{0.5}{a}}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)} \]
19. Add Preprocessing

Alternative 7: 90.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 18.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 18.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 90.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 90.1%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. mul-1-neg 90.1%

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

7. Simplified 90.1%

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

8. Final simplification 90.1%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))