Quadratic roots, medium range

Percentage Accurate: 31.0% → 99.6%

Time: 11.6s

Alternatives: 7

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.0% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. /-rgt-identity 31.7%

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

   2. metadata-eval 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-*r/ 31.7%

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

   5. +-commutative 31.7%

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

   6. unsub-neg 31.7%

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

   7. fma-neg 31.8%

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

   8. associate-*l* 31.8%

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

   9. *-commutative 31.8%

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

   10. distribute-rgt-neg-in 31.8%

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

   11. metadata-eval 31.8%

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

   12. associate-/r* 31.8%

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

   13. metadata-eval 31.8%

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

   14. metadata-eval 31.8%

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

3. Simplified 31.8%

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

   1. fma-udef 31.7%

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

   2. *-commutative 31.7%

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

5. Applied egg-rr 31.7%

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

   1. flip-- 31.7%

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

   2. add-sqr-sqrt 32.8%

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

   3. fma-def 32.6%

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

   4. *-commutative 32.6%

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

   5. fma-def 32.6%

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

   6. *-commutative 32.6%

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

7. Applied egg-rr 32.6%

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

   1. *-commutative 32.6%

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

   2. +-commutative 32.6%

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

   3. metadata-eval 32.6%

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

   4. distribute-lft-neg-in 32.6%

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

   5. fma-neg 32.6%

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

   6. associate-*r* 32.6%

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

   7. *-commutative 32.6%

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

   8. associate-*r* 32.6%

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

   9. fma-neg 32.6%

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

   10. associate-*r* 32.6%

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

   11. *-commutative 32.6%

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

   12. associate-*r* 32.6%

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

   13. distribute-lft-neg-in 32.6%

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

   14. metadata-eval 32.6%

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

   15. *-commutative 32.6%

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

9. Simplified 32.6%

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

10. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

    2. associate-*r* 99.3%

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

12. Simplified 99.3%

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

    1. associate-*l/ 99.4%

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

    2. associate-*r* 99.4%

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

14. Applied egg-rr 99.4%

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

    1. associate-*r/ 99.6%

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

    2. *-commutative 99.6%

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

    3. associate-*r* 99.6%

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

    4. *-commutative 99.6%

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

    5. associate-*r* 99.6%

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

    6. metadata-eval 99.6%

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

16. Simplified 99.6%

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

17. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. /-rgt-identity 31.7%

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

   2. metadata-eval 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-*r/ 31.7%

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

   5. +-commutative 31.7%

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

   6. unsub-neg 31.7%

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

   7. fma-neg 31.8%

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

   8. associate-*l* 31.8%

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

   9. *-commutative 31.8%

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

   10. distribute-rgt-neg-in 31.8%

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

   11. metadata-eval 31.8%

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

   12. associate-/r* 31.8%

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

   13. metadata-eval 31.8%

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

   14. metadata-eval 31.8%

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

3. Simplified 31.8%

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

   1. fma-udef 31.7%

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

   2. *-commutative 31.7%

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

5. Applied egg-rr 31.7%

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

   1. flip-- 31.7%

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

   2. add-sqr-sqrt 32.8%

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

   3. fma-def 32.6%

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

   4. *-commutative 32.6%

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

   5. fma-def 32.6%

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

   6. *-commutative 32.6%

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

7. Applied egg-rr 32.6%

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

   1. *-commutative 32.6%

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

   2. +-commutative 32.6%

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

   3. metadata-eval 32.6%

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

   4. distribute-lft-neg-in 32.6%

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

   5. fma-neg 32.6%

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

   6. associate-*r* 32.6%

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

   7. *-commutative 32.6%

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

   8. associate-*r* 32.6%

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

   9. fma-neg 32.6%

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

   10. associate-*r* 32.6%

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

   11. *-commutative 32.6%

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

   12. associate-*r* 32.6%

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

   13. distribute-lft-neg-in 32.6%

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

   14. metadata-eval 32.6%

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

   15. *-commutative 32.6%

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

9. Simplified 32.6%

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

10. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

    2. associate-*r* 99.3%

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

12. Simplified 99.3%

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

    1. frac-times 99.4%

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

    2. associate-*r* 99.4%

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

14. Applied egg-rr 99.4%

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

    1. *-commutative 99.4%

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

    2. associate-*r* 99.4%

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

    3. *-commutative 99.4%

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

    4. associate-*r* 99.4%

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

    5. metadata-eval 99.4%

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

    6. *-commutative 99.4%

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

16. Simplified 99.4%

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

17. Final simplification 99.4%

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

Alternative 3: 93.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \left(a \cdot \left(a \cdot \frac{c}{\frac{{b}^{3}}{c}} + \frac{c}{b}\right)\right)\right)} \cdot \frac{0.5}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/
   (* c (* a -4.0))
   (+ b (+ b (* -2.0 (* a (+ (* a (/ c (/ (pow b 3.0) c))) (/ c b)))))))
  (/ 0.5 a)))

C

double code(double a, double b, double c) {
	return ((c * (a * -4.0)) / (b + (b + (-2.0 * (a * ((a * (c / (pow(b, 3.0) / c))) + (c / b))))))) * (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 + (b + ((-2.0d0) * (a * ((a * (c / ((b ** 3.0d0) / c))) + (c / b))))))) * (0.5d0 / a)
end function

Java

public static double code(double a, double b, double c) {
	return ((c * (a * -4.0)) / (b + (b + (-2.0 * (a * ((a * (c / (Math.pow(b, 3.0) / c))) + (c / b))))))) * (0.5 / a);
}

Python

def code(a, b, c):
	return ((c * (a * -4.0)) / (b + (b + (-2.0 * (a * ((a * (c / (math.pow(b, 3.0) / c))) + (c / b))))))) * (0.5 / a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(a * -4.0)) / Float64(b + Float64(b + Float64(-2.0 * Float64(a * Float64(Float64(a * Float64(c / Float64((b ^ 3.0) / c))) + Float64(c / b))))))) * Float64(0.5 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * (a * -4.0)) / (b + (b + (-2.0 * (a * ((a * (c / ((b ^ 3.0) / c))) + (c / b))))))) * (0.5 / a);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[(b + N[(-2.0 * N[(a * N[(N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

   1. /-rgt-identity 31.7%

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

   2. metadata-eval 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-*r/ 31.7%

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

   5. +-commutative 31.7%

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

   6. unsub-neg 31.7%

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

   7. fma-neg 31.8%

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

   8. associate-*l* 31.8%

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

   9. *-commutative 31.8%

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

   10. distribute-rgt-neg-in 31.8%

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

   11. metadata-eval 31.8%

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

   12. associate-/r* 31.8%

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

   13. metadata-eval 31.8%

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

   14. metadata-eval 31.8%

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

3. Simplified 31.8%

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

   1. fma-udef 31.7%

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

   2. *-commutative 31.7%

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

5. Applied egg-rr 31.7%

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

   1. flip-- 31.7%

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

   2. add-sqr-sqrt 32.8%

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

   3. fma-def 32.6%

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

   4. *-commutative 32.6%

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

   5. fma-def 32.6%

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

   6. *-commutative 32.6%

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

7. Applied egg-rr 32.6%

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

   1. *-commutative 32.6%

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

   2. +-commutative 32.6%

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

   3. metadata-eval 32.6%

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

   4. distribute-lft-neg-in 32.6%

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

   5. fma-neg 32.6%

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

   6. associate-*r* 32.6%

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

   7. *-commutative 32.6%

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

   8. associate-*r* 32.6%

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

   9. fma-neg 32.6%

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

   10. associate-*r* 32.6%

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

   11. *-commutative 32.6%

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

   12. associate-*r* 32.6%

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

   13. distribute-lft-neg-in 32.6%

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

   14. metadata-eval 32.6%

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

   15. *-commutative 32.6%

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

9. Simplified 32.6%

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

10. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

    2. associate-*r* 99.3%

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

12. Simplified 99.3%

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

13. Taylor expanded in b around inf 92.1%

    \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \color{blue}{\left(-2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(b + -2 \cdot \frac{c \cdot a}{b}\right)\right)}} \cdot \frac{0.5}{a} \]
14. Step-by-step derivation

    1. +-commutative 92.1%

       \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \color{blue}{\left(\left(b + -2 \cdot \frac{c \cdot a}{b}\right) + -2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}} \cdot \frac{0.5}{a} \]

    2. associate-*l/ 92.1%

       \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(\left(b + -2 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)}\right) + -2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)} \cdot \frac{0.5}{a} \]

    3. associate-+l+ 92.1%

       \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \color{blue}{\left(b + \left(-2 \cdot \left(\frac{c}{b} \cdot a\right) + -2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)\right)}} \cdot \frac{0.5}{a} \]

    4. distribute-lft-in 92.1%

       \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + \color{blue}{-2 \cdot \left(\frac{c}{b} \cdot a + \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}\right)}\right)} \cdot \frac{0.5}{a} \]

    5. associate-*l/ 92.1%

       \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \left(\frac{c}{b} \cdot a + \color{blue}{\frac{{c}^{2}}{{b}^{3}} \cdot {a}^{2}}\right)\right)} \cdot \frac{0.5}{a} \]

    6. unpow2 92.1%

       \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \left(\frac{c}{b} \cdot a + \frac{\color{blue}{c \cdot c}}{{b}^{3}} \cdot {a}^{2}\right)\right)} \cdot \frac{0.5}{a} \]

    7. unpow2 92.1%

       \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \left(\frac{c}{b} \cdot a + \frac{c \cdot c}{{b}^{3}} \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)} \cdot \frac{0.5}{a} \]

    8. +-commutative 92.1%

       \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \color{blue}{\left(\frac{c \cdot c}{{b}^{3}} \cdot \left(a \cdot a\right) + \frac{c}{b} \cdot a\right)}\right)} \cdot \frac{0.5}{a} \]

    9. associate-*r* 92.1%

       \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \left(\color{blue}{\left(\frac{c \cdot c}{{b}^{3}} \cdot a\right) \cdot a} + \frac{c}{b} \cdot a\right)\right)} \cdot \frac{0.5}{a} \]

    10. distribute-rgt-out 92.1%

        \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot c}{{b}^{3}} \cdot a + \frac{c}{b}\right)\right)}\right)} \cdot \frac{0.5}{a} \]

    11. associate-/l* 92.1%

        \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \left(a \cdot \left(\color{blue}{\frac{c}{\frac{{b}^{3}}{c}}} \cdot a + \frac{c}{b}\right)\right)\right)} \cdot \frac{0.5}{a} \]

15. Simplified 92.1%

    \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \color{blue}{\left(b + -2 \cdot \left(a \cdot \left(\frac{c}{\frac{{b}^{3}}{c}} \cdot a + \frac{c}{b}\right)\right)\right)}} \cdot \frac{0.5}{a} \]

16. Final simplification 92.1%

    \[\leadsto \frac{c \cdot \left(a \cdot -4\right)}{b + \left(b + -2 \cdot \left(a \cdot \left(a \cdot \frac{c}{\frac{{b}^{3}}{c}} + \frac{c}{b}\right)\right)\right)} \cdot \frac{0.5}{a} \]

Alternative 4: 91.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. neg-sub0 31.7%

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

   2. associate-+l- 31.7%

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

   3. sub0-neg 31.7%

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

   4. neg-mul-1 31.7%

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

   5. associate-*l/ 31.7%

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

   6. *-commutative 31.7%

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

   7. associate-/r* 31.7%

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

   8. /-rgt-identity 31.7%

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

   9. metadata-eval 31.7%

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

3. Simplified 31.7%

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

4. Taylor expanded in b around inf 89.4%

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

   1. +-commutative 89.4%

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

   2. mul-1-neg 89.4%

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

   3. unsub-neg 89.4%

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

   4. associate-*r/ 89.4%

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

   5. neg-mul-1 89.4%

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

   6. *-commutative 89.4%

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

   7. associate-/l* 89.4%

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

   8. unpow2 89.4%

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

6. Simplified 89.4%

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

7. Final simplification 89.4%

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

Alternative 5: 91.0% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. /-rgt-identity 31.7%

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

   2. metadata-eval 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-*r/ 31.7%

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

   5. +-commutative 31.7%

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

   6. unsub-neg 31.7%

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

   7. fma-neg 31.8%

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

   8. associate-*l* 31.8%

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

   9. *-commutative 31.8%

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

   10. distribute-rgt-neg-in 31.8%

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

   11. metadata-eval 31.8%

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

   12. associate-/r* 31.8%

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

   13. metadata-eval 31.8%

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

   14. metadata-eval 31.8%

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

3. Simplified 31.8%

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

   1. fma-udef 31.7%

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

   2. *-commutative 31.7%

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

5. Applied egg-rr 31.7%

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

   1. flip-- 31.7%

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

   2. add-sqr-sqrt 32.8%

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

   3. fma-def 32.6%

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

   4. *-commutative 32.6%

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

   5. fma-def 32.6%

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

   6. *-commutative 32.6%

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

7. Applied egg-rr 32.6%

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

   1. *-commutative 32.6%

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

   2. +-commutative 32.6%

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

   3. metadata-eval 32.6%

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

   4. distribute-lft-neg-in 32.6%

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

   5. fma-neg 32.6%

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

   6. associate-*r* 32.6%

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

   7. *-commutative 32.6%

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

   8. associate-*r* 32.6%

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

   9. fma-neg 32.6%

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

   10. associate-*r* 32.6%

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

   11. *-commutative 32.6%

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

   12. associate-*r* 32.6%

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

   13. distribute-lft-neg-in 32.6%

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

   14. metadata-eval 32.6%

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

   15. *-commutative 32.6%

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

9. Simplified 32.6%

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

10. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

    2. associate-*r* 99.3%

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

12. Simplified 99.3%

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

13. Taylor expanded in b around inf 89.3%

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

14. Final simplification 89.3%

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

Alternative 6: 91.0% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. /-rgt-identity 31.7%

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

   2. metadata-eval 31.7%

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

   3. associate-/l* 31.7%

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

   4. associate-*r/ 31.7%

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

   5. +-commutative 31.7%

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

   6. unsub-neg 31.7%

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

   7. fma-neg 31.8%

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

   8. associate-*l* 31.8%

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

   9. *-commutative 31.8%

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

   10. distribute-rgt-neg-in 31.8%

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

   11. metadata-eval 31.8%

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

   12. associate-/r* 31.8%

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

   13. metadata-eval 31.8%

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

   14. metadata-eval 31.8%

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

3. Simplified 31.8%

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

   1. fma-udef 31.7%

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

   2. *-commutative 31.7%

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

5. Applied egg-rr 31.7%

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

   1. flip-- 31.7%

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

   2. add-sqr-sqrt 32.8%

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

   3. fma-def 32.6%

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

   4. *-commutative 32.6%

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

   5. fma-def 32.6%

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

   6. *-commutative 32.6%

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

7. Applied egg-rr 32.6%

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

   1. *-commutative 32.6%

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

   2. +-commutative 32.6%

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

   3. metadata-eval 32.6%

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

   4. distribute-lft-neg-in 32.6%

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

   5. fma-neg 32.6%

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

   6. associate-*r* 32.6%

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

   7. *-commutative 32.6%

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

   8. associate-*r* 32.6%

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

   9. fma-neg 32.6%

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

   10. associate-*r* 32.6%

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

   11. *-commutative 32.6%

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

   12. associate-*r* 32.6%

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

   13. distribute-lft-neg-in 32.6%

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

   14. metadata-eval 32.6%

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

   15. *-commutative 32.6%

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

9. Simplified 32.6%

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

10. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

    2. associate-*r* 99.3%

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

12. Simplified 99.3%

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

13. Taylor expanded in b around inf 89.3%

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

14. Final simplification 89.3%

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

Alternative 7: 81.7% 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 31.7%

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

   1. neg-sub0 31.7%

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

   2. associate-+l- 31.7%

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

   3. sub0-neg 31.7%

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

   4. neg-mul-1 31.7%

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

   5. associate-*l/ 31.7%

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

   6. *-commutative 31.7%

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

   7. associate-/r* 31.7%

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

   8. /-rgt-identity 31.7%

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

   9. metadata-eval 31.7%

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

3. Simplified 31.7%

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

4. Taylor expanded in b around inf 81.0%

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

   1. associate-*r/ 81.0%

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

   2. neg-mul-1 81.0%

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

6. Simplified 81.0%

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

7. Final simplification 81.0%

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

Reproduce

herbie shell --seed 2023229 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))