Quadratic roots, wide range

Percentage Accurate: 17.9% → 99.9%

Time: 10.6s

Alternatives: 8

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: 17.9% 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

\[\begin{array}{l} \\ \frac{-2 \cdot c}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -8, \mathsf{fma}\left(b, b, c \cdot \left(a \cdot 4\right)\right)\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* -2.0 c) (+ b (sqrt (fma (* c a) -8.0 (fma b b (* c (* a 4.0))))))))

C

double code(double a, double b, double c) {
	return (-2.0 * c) / (b + sqrt(fma((c * a), -8.0, fma(b, b, (c * (a * 4.0))))));
}

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * c) / Float64(b + sqrt(fma(Float64(c * a), -8.0, fma(b, b, Float64(c * Float64(a * 4.0)))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * c), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -8.0 + N[(b * b + N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

   1. *-commutative 17.2%

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

   2. +-commutative 17.2%

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

   3. unsub-neg 17.2%

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

   4. fma-neg 17.3%

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

   5. associate-*l* 17.3%

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

   6. *-commutative 17.3%

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

   7. distribute-rgt-neg-in 17.3%

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

   8. metadata-eval 17.3%

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

3. Simplified 17.3%

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

   1. fma-udef 17.2%

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

   2. *-commutative 17.2%

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

   3. metadata-eval 17.2%

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

   4. cancel-sign-sub-inv 17.2%

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

   5. associate-*l* 17.2%

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

   6. *-un-lft-identity 17.2%

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

   7. prod-diff 17.3%

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

5. Applied egg-rr 17.2%

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

   1. +-commutative 17.2%

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

   2. fma-udef 17.2%

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

   3. *-rgt-identity 17.2%

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

   4. *-rgt-identity 17.2%

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

   5. count-2 17.2%

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

   6. *-commutative 17.2%

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

   7. *-commutative 17.2%

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

   8. associate-*r* 17.2%

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

   9. *-rgt-identity 17.2%

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

   10. fma-neg 17.2%

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

   11. *-commutative 17.2%

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

   12. *-commutative 17.2%

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

   13. associate-*r* 17.2%

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

7. Simplified 17.2%

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

   1. flip-- 17.1%

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

   2. add-sqr-sqrt 17.5%

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

   3. associate-*r* 17.5%

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

   4. metadata-eval 17.5%

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

   5. fma-neg 17.5%

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

   6. *-commutative 17.5%

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

   7. *-commutative 17.5%

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

   8. associate-*r* 17.5%

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

9. Applied egg-rr 17.5%

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

10. Taylor expanded in a around 0 99.4%

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

    1. div-inv 99.3%

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

    2. distribute-rgt-out-- 99.3%

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

    3. metadata-eval 99.3%

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

    4. *-commutative 99.3%

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

    5. associate-*l* 99.3%

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

    6. +-commutative 99.3%

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

    7. fma-def 99.3%

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

    8. distribute-rgt-neg-in 99.3%

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

    9. *-commutative 99.3%

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

12. Applied egg-rr 99.3%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{a \cdot \left(c \cdot -8 - c \cdot -4\right)}{b + \sqrt{\left(c \cdot a\right) \cdot -8 + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot \left(--4\right)\right)\right)}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (* a (- (* c -8.0) (* c -4.0)))
   (+ b (sqrt (+ (* (* c a) -8.0) (fma b b (* a (* c (- -4.0))))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return ((a * ((c * -8.0) - (c * -4.0))) / (b + sqrt((((c * a) * -8.0) + fma(b, b, (a * (c * -(-4.0)))))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(Float64(c * -8.0) - Float64(c * -4.0))) / Float64(b + sqrt(Float64(Float64(Float64(c * a) * -8.0) + fma(b, b, Float64(a * Float64(c * Float64(-(-4.0))))))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(N[(c * -8.0), $MachinePrecision] - N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(N[(c * a), $MachinePrecision] * -8.0), $MachinePrecision] + N[(b * b + N[(a * N[(c * (--4.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

   1. *-commutative 17.2%

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

   2. +-commutative 17.2%

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

   3. unsub-neg 17.2%

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

   4. fma-neg 17.3%

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

   5. associate-*l* 17.3%

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

   6. *-commutative 17.3%

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

   7. distribute-rgt-neg-in 17.3%

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

   8. metadata-eval 17.3%

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

3. Simplified 17.3%

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

   1. fma-udef 17.2%

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

   2. *-commutative 17.2%

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

   3. metadata-eval 17.2%

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

   4. cancel-sign-sub-inv 17.2%

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

   5. associate-*l* 17.2%

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

   6. *-un-lft-identity 17.2%

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

   7. prod-diff 17.3%

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

5. Applied egg-rr 17.2%

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

   1. +-commutative 17.2%

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

   2. fma-udef 17.2%

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

   3. *-rgt-identity 17.2%

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

   4. *-rgt-identity 17.2%

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

   5. count-2 17.2%

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

   6. *-commutative 17.2%

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

   7. *-commutative 17.2%

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

   8. associate-*r* 17.2%

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

   9. *-rgt-identity 17.2%

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

   10. fma-neg 17.2%

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

   11. *-commutative 17.2%

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

   12. *-commutative 17.2%

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

   13. associate-*r* 17.2%

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

7. Simplified 17.2%

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

   1. flip-- 17.1%

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

   2. add-sqr-sqrt 17.5%

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

   3. associate-*r* 17.5%

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

   4. metadata-eval 17.5%

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

   5. fma-neg 17.5%

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

   6. *-commutative 17.5%

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

   7. *-commutative 17.5%

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

   8. associate-*r* 17.5%

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

9. Applied egg-rr 17.5%

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

10. Taylor expanded in a around 0 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -8 - c \cdot -4\right)\\ \frac{\frac{t_0}{b + \sqrt{t_0 + {b}^{2}}}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (- (* c -8.0) (* c -4.0)))))
   (/ (/ t_0 (+ b (sqrt (+ t_0 (pow b 2.0))))) (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = a * ((c * -8.0) - (c * -4.0));
	return (t_0 / (b + sqrt((t_0 + pow(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
    real(8) :: t_0
    t_0 = a * ((c * (-8.0d0)) - (c * (-4.0d0)))
    code = (t_0 / (b + sqrt((t_0 + (b ** 2.0d0))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * ((c * -8.0) - (c * -4.0));
	return (t_0 / (b + Math.sqrt((t_0 + Math.pow(b, 2.0))))) / (a * 2.0);
}

Python

def code(a, b, c):
	t_0 = a * ((c * -8.0) - (c * -4.0))
	return (t_0 / (b + math.sqrt((t_0 + math.pow(b, 2.0))))) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(Float64(c * -8.0) - Float64(c * -4.0)))
	return Float64(Float64(t_0 / Float64(b + sqrt(Float64(t_0 + (b ^ 2.0))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * ((c * -8.0) - (c * -4.0));
	tmp = (t_0 / (b + sqrt((t_0 + (b ^ 2.0))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(N[(c * -8.0), $MachinePrecision] - N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(b + N[Sqrt[N[(t$95$0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 17.2%

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

   1. *-commutative 17.2%

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

   2. +-commutative 17.2%

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

   3. unsub-neg 17.2%

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

   4. fma-neg 17.3%

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

   5. associate-*l* 17.3%

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

   6. *-commutative 17.3%

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

   7. distribute-rgt-neg-in 17.3%

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

   8. metadata-eval 17.3%

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

3. Simplified 17.3%

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

   1. fma-udef 17.2%

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

   2. *-commutative 17.2%

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

   3. metadata-eval 17.2%

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

   4. cancel-sign-sub-inv 17.2%

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

   5. associate-*l* 17.2%

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

   6. *-un-lft-identity 17.2%

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

   7. prod-diff 17.3%

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

5. Applied egg-rr 17.2%

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

   1. +-commutative 17.2%

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

   2. fma-udef 17.2%

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

   3. *-rgt-identity 17.2%

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

   4. *-rgt-identity 17.2%

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

   5. count-2 17.2%

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

   6. *-commutative 17.2%

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

   7. *-commutative 17.2%

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

   8. associate-*r* 17.2%

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

   9. *-rgt-identity 17.2%

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

   10. fma-neg 17.2%

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

   11. *-commutative 17.2%

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

   12. *-commutative 17.2%

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

   13. associate-*r* 17.2%

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

7. Simplified 17.2%

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

   1. flip-- 17.1%

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

   2. add-sqr-sqrt 17.5%

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

   3. associate-*r* 17.5%

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

   4. metadata-eval 17.5%

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

   5. fma-neg 17.5%

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

   6. *-commutative 17.5%

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

   7. *-commutative 17.5%

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

   8. associate-*r* 17.5%

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

9. Applied egg-rr 17.5%

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

10. Taylor expanded in a around 0 99.4%

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

11. Taylor expanded in a around 0 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.2%

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

   1. /-rgt-identity 17.2%

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

   2. metadata-eval 17.2%

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

   3. associate-/l* 17.2%

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

   4. associate-*r/ 17.2%

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

   5. +-commutative 17.2%

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

   6. unsub-neg 17.2%

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

   7. fma-neg 17.3%

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

   8. associate-*l* 17.3%

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

   9. *-commutative 17.3%

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

   10. distribute-rgt-neg-in 17.3%

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

   11. metadata-eval 17.3%

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

   12. associate-/r* 17.3%

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

   13. metadata-eval 17.3%

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

   14. metadata-eval 17.3%

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

3. Simplified 17.3%

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

   1. fma-udef 17.2%

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

   2. *-commutative 17.2%

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

   3. metadata-eval 17.2%

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

   4. cancel-sign-sub-inv 17.2%

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

   5. associate-*l* 17.2%

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

   6. *-un-lft-identity 17.2%

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

   7. prod-diff 17.3%

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

5. Applied egg-rr 17.1%

   \[\leadsto \left(\sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]
6. Step-by-step derivation

   1. *-rgt-identity 17.1%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, -\color{blue}{a \cdot \left(c \cdot -4\right)}\right) + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   2. fma-neg 17.1%

      \[\leadsto \left(\sqrt{\color{blue}{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right)} + \mathsf{fma}\left(a \cdot \left(c \cdot -4\right), 1, \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   3. fma-udef 17.1%

      \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \color{blue}{\left(\left(a \cdot \left(c \cdot -4\right)\right) \cdot 1 + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)}} - b\right) \cdot \frac{0.5}{a} \]

   4. *-rgt-identity 17.1%

      \[\leadsto \left(\sqrt{\left(b \cdot b - a \cdot \left(c \cdot -4\right)\right) + \left(\color{blue}{a \cdot \left(c \cdot -4\right)} + \left(a \cdot \left(c \cdot -4\right)\right) \cdot 1\right)} - b\right) \cdot \frac{0.5}{a} \]

   5. *-rgt-identity 17.1%

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

   6. associate--r- 17.2%

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

   7. associate--r+ 17.2%

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

   8. +-inverses 17.2%

      \[\leadsto \left(\sqrt{b \cdot b - \left(\color{blue}{0} - a \cdot \left(c \cdot -4\right)\right)} - b\right) \cdot \frac{0.5}{a} \]

   9. neg-sub0 17.2%

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

   10. associate-*r* 17.2%

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

   11. distribute-rgt-neg-in 17.2%

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

   12. metadata-eval 17.2%

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

   13. *-commutative 17.2%

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

   14. associate-*r* 17.2%

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

7. Simplified 17.2%

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

   1. flip-- 17.2%

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

   2. add-sqr-sqrt 17.7%

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

   3. *-commutative 17.7%

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

   4. *-commutative 17.7%

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

9. Applied egg-rr 17.7%

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

10. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 5: 95.2% 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 17.2%

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

   1. /-rgt-identity 17.2%

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

   2. metadata-eval 17.2%

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

   3. associate-/l* 17.2%

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

   4. associate-*r/ 17.2%

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

   5. +-commutative 17.2%

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

   6. unsub-neg 17.2%

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

   7. fma-neg 17.3%

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

   8. associate-*l* 17.3%

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

   9. *-commutative 17.3%

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

   10. distribute-rgt-neg-in 17.3%

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

   11. metadata-eval 17.3%

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

   12. associate-/r* 17.3%

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

   13. metadata-eval 17.3%

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

   14. metadata-eval 17.3%

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

3. Simplified 17.3%

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

4. Taylor expanded in b around inf 95.6%

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

   1. +-commutative 95.6%

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

   2. mul-1-neg 95.6%

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

   3. unsub-neg 95.6%

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

   4. mul-1-neg 95.6%

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

   5. distribute-neg-frac 95.6%

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

   6. associate-/l* 95.6%

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

   7. unpow2 95.6%

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

6. Simplified 95.6%

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

7. Final simplification 95.6%

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

Alternative 6: 94.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -8 - c \cdot -4\right)\\ \frac{\frac{t_0}{0.5 \cdot \frac{t_0}{b} + b \cdot 2}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (- (* c -8.0) (* c -4.0)))))
   (/ (/ t_0 (+ (* 0.5 (/ t_0 b)) (* b 2.0))) (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = a * ((c * -8.0) - (c * -4.0));
	return (t_0 / ((0.5 * (t_0 / 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
    real(8) :: t_0
    t_0 = a * ((c * (-8.0d0)) - (c * (-4.0d0)))
    code = (t_0 / ((0.5d0 * (t_0 / b)) + (b * 2.0d0))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * ((c * -8.0) - (c * -4.0));
	return (t_0 / ((0.5 * (t_0 / b)) + (b * 2.0))) / (a * 2.0);
}

Python

def code(a, b, c):
	t_0 = a * ((c * -8.0) - (c * -4.0))
	return (t_0 / ((0.5 * (t_0 / b)) + (b * 2.0))) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(Float64(c * -8.0) - Float64(c * -4.0)))
	return Float64(Float64(t_0 / Float64(Float64(0.5 * Float64(t_0 / b)) + Float64(b * 2.0))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * ((c * -8.0) - (c * -4.0));
	tmp = (t_0 / ((0.5 * (t_0 / b)) + (b * 2.0))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(N[(c * -8.0), $MachinePrecision] - N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(N[(0.5 * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 17.2%

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

   1. *-commutative 17.2%

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

   2. +-commutative 17.2%

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

   3. unsub-neg 17.2%

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

   4. fma-neg 17.3%

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

   5. associate-*l* 17.3%

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

   6. *-commutative 17.3%

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

   7. distribute-rgt-neg-in 17.3%

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

   8. metadata-eval 17.3%

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

3. Simplified 17.3%

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

   1. fma-udef 17.2%

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

   2. *-commutative 17.2%

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

   3. metadata-eval 17.2%

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

   4. cancel-sign-sub-inv 17.2%

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

   5. associate-*l* 17.2%

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

   6. *-un-lft-identity 17.2%

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

   7. prod-diff 17.3%

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

5. Applied egg-rr 17.2%

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

   1. +-commutative 17.2%

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

   2. fma-udef 17.2%

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

   3. *-rgt-identity 17.2%

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

   4. *-rgt-identity 17.2%

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

   5. count-2 17.2%

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

   6. *-commutative 17.2%

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

   7. *-commutative 17.2%

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

   8. associate-*r* 17.2%

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

   9. *-rgt-identity 17.2%

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

   10. fma-neg 17.2%

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

   11. *-commutative 17.2%

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

   12. *-commutative 17.2%

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

   13. associate-*r* 17.2%

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

7. Simplified 17.2%

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

   1. flip-- 17.1%

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

   2. add-sqr-sqrt 17.5%

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

   3. associate-*r* 17.5%

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

   4. metadata-eval 17.5%

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

   5. fma-neg 17.5%

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

   6. *-commutative 17.5%

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

   7. *-commutative 17.5%

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

   8. associate-*r* 17.5%

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

9. Applied egg-rr 17.5%

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

10. Taylor expanded in a around 0 99.4%

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

11. Taylor expanded in a around 0 95.3%

    \[\leadsto \frac{\frac{\left(-8 \cdot c - -4 \cdot c\right) \cdot a}{\color{blue}{0.5 \cdot \frac{\left(-8 \cdot c - -4 \cdot c\right) \cdot a}{b} + 2 \cdot b}}}{a \cdot 2} \]

12. Final simplification 95.3%

    \[\leadsto \frac{\frac{a \cdot \left(c \cdot -8 - c \cdot -4\right)}{0.5 \cdot \frac{a \cdot \left(c \cdot -8 - c \cdot -4\right)}{b} + b \cdot 2}}{a \cdot 2} \]

Alternative 7: 94.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.2%

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

   1. *-commutative 17.2%

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

   2. +-commutative 17.2%

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

   3. unsub-neg 17.2%

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

   4. fma-neg 17.3%

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

   5. associate-*l* 17.3%

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

   6. *-commutative 17.3%

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

   7. distribute-rgt-neg-in 17.3%

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

   8. metadata-eval 17.3%

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

3. Simplified 17.3%

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

4. Taylor expanded in b around inf 12.5%

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

   1. associate-*r/ 12.5%

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

6. Simplified 12.5%

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

   1. flip-- 12.5%

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

   2. associate-/l* 12.5%

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

   3. associate-/l* 12.5%

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

   4. associate-/l* 12.5%

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

8. Applied egg-rr 12.5%

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

9. Taylor expanded in b around inf 95.3%

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

    1. *-commutative 95.3%

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

11. Simplified 95.3%

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

12. Final simplification 95.3%

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

Alternative 8: 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(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 17.2%

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

   1. /-rgt-identity 17.2%

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

   2. metadata-eval 17.2%

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

   3. associate-/l* 17.2%

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

   4. associate-*r/ 17.2%

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

   5. +-commutative 17.2%

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

   6. unsub-neg 17.2%

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

   7. fma-neg 17.3%

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

   8. associate-*l* 17.3%

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

   9. *-commutative 17.3%

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

   10. distribute-rgt-neg-in 17.3%

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

   11. metadata-eval 17.3%

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

   12. associate-/r* 17.3%

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

   13. metadata-eval 17.3%

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

   14. metadata-eval 17.3%

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

3. Simplified 17.3%

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

4. Taylor expanded in b around inf 90.9%

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

   1. mul-1-neg 90.9%

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

   2. distribute-neg-frac 90.9%

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

6. Simplified 90.9%

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

7. Final simplification 90.9%

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

Reproduce

herbie shell --seed 2023217 
(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)))