Quadratic roots, wide range

Percentage Accurate: 17.6% → 99.5%

Time: 9.9s

Alternatives: 6

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: 17.6% 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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * (a * -4.0)) * (0.5 / a)) / (b + sqrt(fma(c, (a * -4.0), (b * b))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. *-commutative 16.9%

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

   2. +-commutative 16.9%

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

   3. unsub-neg 16.9%

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

   4. fma-neg 17.0%

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

   5. associate-*l* 17.0%

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

   6. *-commutative 17.0%

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

   7. distribute-rgt-neg-in 17.0%

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

   8. metadata-eval 17.0%

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

3. Simplified 17.0%

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

   1. fma-udef 16.9%

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

   2. *-commutative 16.9%

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

   3. metadata-eval 16.9%

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

   4. cancel-sign-sub-inv 16.9%

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

   5. associate-*l* 16.9%

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

   6. *-un-lft-identity 16.9%

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

   7. prod-diff 17.0%

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

5. Applied egg-rr 16.9%

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

   1. +-commutative 16.9%

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

   2. fma-udef 16.9%

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

   3. *-rgt-identity 16.9%

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

   4. *-rgt-identity 16.9%

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

   5. count-2 16.9%

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

   6. *-commutative 16.9%

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

   7. *-commutative 16.9%

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

   8. associate-*r* 16.9%

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

   9. *-rgt-identity 16.9%

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

   10. fma-neg 16.9%

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

   11. *-commutative 16.9%

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

   12. *-commutative 16.9%

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

   13. associate-*r* 16.9%

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

7. Simplified 16.9%

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

   1. flip-- 16.9%

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

   2. add-sqr-sqrt 17.2%

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

   3. associate-*r* 17.2%

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

   4. metadata-eval 17.2%

      \[\leadsto \frac{\frac{\left(\color{blue}{-8} \cdot \left(c \cdot a\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   5. cancel-sign-sub-inv 17.2%

      \[\leadsto \frac{\frac{\left(-8 \cdot \left(c \cdot a\right) + \color{blue}{\left(b \cdot b + \left(--4\right) \cdot \left(c \cdot a\right)\right)}\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   6. metadata-eval 17.2%

      \[\leadsto \frac{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + \color{blue}{4} \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

9. Applied egg-rr 17.2%

   \[\leadsto \frac{\color{blue}{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}}}{a \cdot 2} \]
10. Step-by-step derivation

11. Simplified 99.5%

    \[\leadsto \frac{\color{blue}{\frac{\left(c \cdot a\right) \cdot -4 + 0 \cdot \left(b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]
12. Step-by-step derivation

    1. div-inv 99.3%

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

    2. fma-def 99.3%

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

    3. mul0-lft 99.3%

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

13. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{1}{a \cdot 2}} \]
14. Step-by-step derivation

    1. *-lft-identity 99.3%

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

    2. associate-*r/ 99.3%

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

    3. associate-*l/ 99.5%

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

    4. fma-udef 99.5%

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

    5. metadata-eval 99.5%

       \[\leadsto \frac{\left(1 \cdot \left(\left(c \cdot a\right) \cdot \color{blue}{\left(-8 + 4\right)} + 0\right)\right) \cdot \frac{1}{a \cdot 2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]

    6. distribute-rgt-out 99.5%

       \[\leadsto \frac{\left(1 \cdot \left(\color{blue}{\left(-8 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right)} + 0\right)\right) \cdot \frac{1}{a \cdot 2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]

    7. +-rgt-identity 99.5%

       \[\leadsto \frac{\left(1 \cdot \color{blue}{\left(-8 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \frac{1}{a \cdot 2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]

    8. *-lft-identity 99.5%

       \[\leadsto \frac{\color{blue}{\left(-8 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right)} \cdot \frac{1}{a \cdot 2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]

    9. distribute-rgt-out 99.5%

       \[\leadsto \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot \left(-8 + 4\right)\right)} \cdot \frac{1}{a \cdot 2}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \]

    10. metadata-eval 99.5%

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

    11. associate-*r* 99.5%

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

    12. *-commutative 99.5%

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

    13. associate-/r* 99.5%

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

    14. metadata-eval 99.5%

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

15. Simplified 99.5%

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

16. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right) \cdot 0}{b + \sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (+ (* -4.0 (* c a)) (* (* b b) 0.0))
   (+ b (sqrt (+ (* c (* a -4.0)) (* b b)))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (((-4.0 * (c * a)) + ((b * b) * 0.0)) / (b + sqrt(((c * (a * -4.0)) + (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 = ((((-4.0d0) * (c * a)) + ((b * b) * 0.0d0)) / (b + sqrt(((c * (a * (-4.0d0))) + (b * b))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (((-4.0 * (c * a)) + ((b * b) * 0.0)) / (b + Math.sqrt(((c * (a * -4.0)) + (b * b))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (((-4.0 * (c * a)) + ((b * b) * 0.0)) / (b + math.sqrt(((c * (a * -4.0)) + (b * b))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(-4.0 * Float64(c * a)) + Float64(Float64(b * b) * 0.0)) / Float64(b + sqrt(Float64(Float64(c * Float64(a * -4.0)) + Float64(b * b))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((-4.0 * (c * a)) + ((b * b) * 0.0)) / (b + sqrt(((c * (a * -4.0)) + (b * b))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.9%

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

   1. *-commutative 16.9%

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

   2. +-commutative 16.9%

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

   3. unsub-neg 16.9%

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

   4. fma-neg 17.0%

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

   5. associate-*l* 17.0%

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

   6. *-commutative 17.0%

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

   7. distribute-rgt-neg-in 17.0%

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

   8. metadata-eval 17.0%

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

3. Simplified 17.0%

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

   1. fma-udef 16.9%

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

   2. *-commutative 16.9%

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

   3. metadata-eval 16.9%

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

   4. cancel-sign-sub-inv 16.9%

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

   5. associate-*l* 16.9%

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

   6. *-un-lft-identity 16.9%

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

   7. prod-diff 17.0%

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

5. Applied egg-rr 16.9%

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

   1. +-commutative 16.9%

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

   2. fma-udef 16.9%

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

   3. *-rgt-identity 16.9%

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

   4. *-rgt-identity 16.9%

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

   5. count-2 16.9%

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

   6. *-commutative 16.9%

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

   7. *-commutative 16.9%

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

   8. associate-*r* 16.9%

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

   9. *-rgt-identity 16.9%

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

   10. fma-neg 16.9%

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

   11. *-commutative 16.9%

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

   12. *-commutative 16.9%

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

   13. associate-*r* 16.9%

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

7. Simplified 16.9%

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

   1. flip-- 16.9%

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

   2. add-sqr-sqrt 17.2%

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

   3. associate-*r* 17.2%

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

   4. metadata-eval 17.2%

      \[\leadsto \frac{\frac{\left(\color{blue}{-8} \cdot \left(c \cdot a\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   5. cancel-sign-sub-inv 17.2%

      \[\leadsto \frac{\frac{\left(-8 \cdot \left(c \cdot a\right) + \color{blue}{\left(b \cdot b + \left(--4\right) \cdot \left(c \cdot a\right)\right)}\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   6. metadata-eval 17.2%

      \[\leadsto \frac{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + \color{blue}{4} \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

9. Applied egg-rr 17.2%

   \[\leadsto \frac{\color{blue}{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}}}{a \cdot 2} \]
10. Step-by-step derivation

11. Simplified 99.5%

    \[\leadsto \frac{\color{blue}{\frac{\left(c \cdot a\right) \cdot -4 + 0 \cdot \left(b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}}{a \cdot 2} \]
12. Step-by-step derivation

    1. fma-udef 99.5%

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

13. Applied egg-rr 99.5%

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

14. Final simplification 99.5%

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

Alternative 3: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. /-rgt-identity 16.9%

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

   2. metadata-eval 16.9%

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

   3. associate-/l* 16.9%

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

   4. associate-*r/ 16.9%

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

   5. +-commutative 16.9%

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

   6. unsub-neg 16.9%

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

   7. fma-neg 17.0%

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

   8. associate-*l* 17.0%

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

   9. *-commutative 17.0%

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

   10. distribute-rgt-neg-in 17.0%

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

   11. metadata-eval 17.0%

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

   12. associate-/r* 17.0%

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

   13. metadata-eval 17.0%

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

   14. metadata-eval 17.0%

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

3. Simplified 17.0%

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

4. Taylor expanded in b around inf 96.3%

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

   1. +-commutative 96.3%

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

   2. mul-1-neg 96.3%

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

   3. unsub-neg 96.3%

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

   4. mul-1-neg 96.3%

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

   5. distribute-neg-frac 96.3%

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

   6. associate-/l* 96.3%

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

   7. unpow2 96.3%

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

6. Simplified 96.3%

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

7. Final simplification 96.3%

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

Alternative 4: 95.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right) \cdot 0}{-2 \cdot \frac{c \cdot a}{b} + b \cdot 2}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (+ (* -4.0 (* c a)) (* (* b b) 0.0)) (+ (* -2.0 (/ (* c a) b)) (* b 2.0)))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (((-4.0 * (c * a)) + ((b * b) * 0.0)) / ((-2.0 * ((c * a) / b)) + (b * 2.0))) / (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 = ((((-4.0d0) * (c * a)) + ((b * b) * 0.0d0)) / (((-2.0d0) * ((c * a) / b)) + (b * 2.0d0))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (((-4.0 * (c * a)) + ((b * b) * 0.0)) / ((-2.0 * ((c * a) / b)) + (b * 2.0))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (((-4.0 * (c * a)) + ((b * b) * 0.0)) / ((-2.0 * ((c * a) / b)) + (b * 2.0))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(-4.0 * Float64(c * a)) + Float64(Float64(b * b) * 0.0)) / Float64(Float64(-2.0 * Float64(Float64(c * a) / b)) + Float64(b * 2.0))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((-4.0 * (c * a)) + ((b * b) * 0.0)) / ((-2.0 * ((c * a) / b)) + (b * 2.0))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.9%

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

   1. *-commutative 16.9%

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

   2. +-commutative 16.9%

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

   3. unsub-neg 16.9%

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

   4. fma-neg 17.0%

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

   5. associate-*l* 17.0%

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

   6. *-commutative 17.0%

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

   7. distribute-rgt-neg-in 17.0%

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

   8. metadata-eval 17.0%

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

3. Simplified 17.0%

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

   1. fma-udef 16.9%

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

   2. *-commutative 16.9%

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

   3. metadata-eval 16.9%

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

   4. cancel-sign-sub-inv 16.9%

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

   5. associate-*l* 16.9%

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

   6. *-un-lft-identity 16.9%

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

   7. prod-diff 17.0%

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

5. Applied egg-rr 16.9%

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

   1. +-commutative 16.9%

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

   2. fma-udef 16.9%

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

   3. *-rgt-identity 16.9%

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

   4. *-rgt-identity 16.9%

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

   5. count-2 16.9%

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

   6. *-commutative 16.9%

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

   7. *-commutative 16.9%

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

   8. associate-*r* 16.9%

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

   9. *-rgt-identity 16.9%

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

   10. fma-neg 16.9%

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

   11. *-commutative 16.9%

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

   12. *-commutative 16.9%

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

   13. associate-*r* 16.9%

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

7. Simplified 16.9%

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

   1. flip-- 16.9%

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

   2. add-sqr-sqrt 17.2%

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

   3. associate-*r* 17.2%

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

   4. metadata-eval 17.2%

      \[\leadsto \frac{\frac{\left(\color{blue}{-8} \cdot \left(c \cdot a\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   5. cancel-sign-sub-inv 17.2%

      \[\leadsto \frac{\frac{\left(-8 \cdot \left(c \cdot a\right) + \color{blue}{\left(b \cdot b + \left(--4\right) \cdot \left(c \cdot a\right)\right)}\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

   6. metadata-eval 17.2%

      \[\leadsto \frac{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + \color{blue}{4} \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot b - -4 \cdot \left(c \cdot a\right)\right)} + b}}{a \cdot 2} \]

9. Applied egg-rr 17.2%

   \[\leadsto \frac{\color{blue}{\frac{\left(-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)\right) - b \cdot b}{\sqrt{-8 \cdot \left(c \cdot a\right) + \left(b \cdot b + 4 \cdot \left(c \cdot a\right)\right)} + b}}}{a \cdot 2} \]
10. Step-by-step derivation

11. Simplified 99.5%

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

12. Taylor expanded in b around inf 96.0%

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

13. Final simplification 96.0%

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

Alternative 5: 95.2% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (0.5d0 / a) * (((-4.0d0) * (c * a)) / (b + (b + ((-2.0d0) * (c / (b / a))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. /-rgt-identity 16.9%

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

   2. metadata-eval 16.9%

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

   3. associate-/l* 16.9%

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

   4. associate-*r/ 16.9%

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

   5. +-commutative 16.9%

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

   6. unsub-neg 16.9%

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

   7. fma-neg 17.0%

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

   8. associate-*l* 17.0%

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

   9. *-commutative 17.0%

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

   10. distribute-rgt-neg-in 17.0%

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

   11. metadata-eval 17.0%

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

   12. associate-/r* 17.0%

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

   13. metadata-eval 17.0%

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

   14. metadata-eval 17.0%

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

3. Simplified 17.0%

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

4. Taylor expanded in b around inf 12.9%

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

   1. associate-*r/ 12.9%

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

6. Simplified 12.9%

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

   1. flip-- 12.9%

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

   2. associate-*r/ 12.9%

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

   3. associate-/l* 12.9%

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

   4. associate-*r/ 12.9%

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

   5. associate-/l* 12.9%

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

   6. associate-*r/ 12.9%

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

   7. associate-/l* 12.9%

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

8. Applied egg-rr 12.9%

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

9. Taylor expanded in b around inf 95.8%

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

    1. *-commutative 95.8%

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

11. Simplified 95.8%

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

12. Final simplification 95.8%

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

Alternative 6: 90.6% 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(Float64(-c) / 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 16.9%

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

   1. /-rgt-identity 16.9%

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

   2. metadata-eval 16.9%

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

   3. associate-/l* 16.9%

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

   4. associate-*r/ 16.9%

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

   5. +-commutative 16.9%

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

   6. unsub-neg 16.9%

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

   7. fma-neg 17.0%

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

   8. associate-*l* 17.0%

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

   9. *-commutative 17.0%

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

   10. distribute-rgt-neg-in 17.0%

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

   11. metadata-eval 17.0%

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

   12. associate-/r* 17.0%

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

   13. metadata-eval 17.0%

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

   14. metadata-eval 17.0%

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

3. Simplified 17.0%

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

4. Taylor expanded in b around inf 91.2%

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

   1. mul-1-neg 91.2%

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

   2. distribute-neg-frac 91.2%

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

6. Simplified 91.2%

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

7. Final simplification 91.2%

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

Reproduce

herbie shell --seed 2023200 
(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)))