Quadratic roots, wide range

Percentage Accurate: 17.8% → 99.7%

Time: 49.3s

Alternatives: 5

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: 17.8% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

   1. add-exp-log 16.8%

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

   2. associate-*l* 16.8%

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

6. Applied egg-rr 16.8%

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

   1. flip-+ 16.7%

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

   2. pow2 16.7%

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

   3. add-sqr-sqrt 17.1%

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

   4. pow2 17.1%

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

   5. rem-exp-log 17.2%

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

   6. associate-*r* 17.2%

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

   7. pow2 17.2%

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

   8. rem-exp-log 17.2%

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

   9. associate-*r* 17.2%

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

8. Applied egg-rr 17.2%

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

9. Taylor expanded in b around 0 99.5%

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

    1. *-commutative 99.5%

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

    2. associate-*l* 99.5%

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

11. Simplified 99.5%

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

    1. *-un-lft-identity 99.5%

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

    2. associate-/l/ 99.4%

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

    3. *-commutative 99.4%

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

    4. *-commutative 99.4%

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

13. Applied egg-rr 99.4%

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

    1. *-lft-identity 99.4%

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

    2. associate-/r* 99.7%

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

    3. associate-*r* 99.7%

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

    4. *-commutative 99.7%

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

    5. associate-*r* 99.7%

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

    6. unpow2 99.7%

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

    7. associate-*r* 99.7%

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

    8. *-commutative 99.7%

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

    9. associate-*r* 99.7%

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

    10. fma-neg 99.7%

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

    11. associate-*r* 99.7%

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

    12. distribute-rgt-neg-in 99.7%

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

    13. metadata-eval 99.7%

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

15. Simplified 99.7%

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

16. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((a * (c * 4.0d0)) / (-b - sqrt(((b * b) - (c * (a * 4.0d0)))))) / (a * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

   1. add-exp-log 16.8%

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

   2. associate-*l* 16.8%

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

6. Applied egg-rr 16.8%

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

   1. flip-+ 16.7%

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

   2. pow2 16.7%

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

   3. add-sqr-sqrt 17.1%

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

   4. pow2 17.1%

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

   5. rem-exp-log 17.2%

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

   6. associate-*r* 17.2%

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

   7. pow2 17.2%

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

   8. rem-exp-log 17.2%

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

   9. associate-*r* 17.2%

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

8. Applied egg-rr 17.2%

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

9. Taylor expanded in b around 0 99.5%

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

    1. *-commutative 99.5%

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

    2. associate-*l* 99.5%

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

11. Simplified 99.5%

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

    1. unpow2 99.5%

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

13. Applied egg-rr 99.5%

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

14. Final simplification 99.5%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 97.2%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-1 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-2 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{c \cdot \left(4 \cdot \frac{{a}^{4}}{{b}^{6}} + 16 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - \frac{1}{b}\right)} \]
6. Step-by-step derivation

7. Simplified 97.2%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-1, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 20}{a \cdot b}\right)\right)\right) - \frac{1}{b}\right)} \]

8. Taylor expanded in b around inf 95.1%

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

   1. neg-mul-1 95.1%

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

   2. mul-1-neg 95.1%

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

   3. unsub-neg 95.1%

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

   4. associate-/l* 95.1%

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

   5. unpow2 95.1%

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

   6. unpow2 95.1%

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

   7. times-frac 95.1%

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

   8. sqr-neg 95.1%

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

   9. distribute-frac-neg 95.1%

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

   10. distribute-frac-neg 95.1%

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

   11. unpow1 95.1%

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

   12. pow-plus 95.1%

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

   13. distribute-frac-neg 95.1%

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

   14. distribute-neg-frac2 95.1%

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

   15. metadata-eval 95.1%

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

10. Simplified 95.1%

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

Alternative 4: 95.1% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((a * (c * 4.0d0)) / (2.0d0 * ((a * (c / b)) - b))) / (a * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

   1. add-exp-log 16.8%

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

   2. associate-*l* 16.8%

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

6. Applied egg-rr 16.8%

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

   1. flip-+ 16.7%

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

   2. pow2 16.7%

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

   3. add-sqr-sqrt 17.1%

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

   4. pow2 17.1%

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

   5. rem-exp-log 17.2%

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

   6. associate-*r* 17.2%

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

   7. pow2 17.2%

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

   8. rem-exp-log 17.2%

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

   9. associate-*r* 17.2%

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

8. Applied egg-rr 17.2%

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

9. Taylor expanded in b around 0 99.5%

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

    1. *-commutative 99.5%

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

    2. associate-*l* 99.5%

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

11. Simplified 99.5%

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

12. Taylor expanded in a around 0 94.9%

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

    1. distribute-lft-out-- 94.9%

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

    2. associate-/l* 94.9%

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

14. Simplified 94.9%

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

Alternative 5: 90.4% 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 16.8%

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

   1. *-commutative 16.8%

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

3. Simplified 16.8%

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

5. Taylor expanded in b around inf 91.0%

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

   1. associate-*r/ 91.0%

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

   2. mul-1-neg 91.0%

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

7. Simplified 91.0%

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

8. Final simplification 91.0%

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

Reproduce

herbie shell --seed 2024116 
(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)))