Quadratic roots, medium range

Percentage Accurate: 31.5% → 99.6%

Time: 16.2s

Alternatives: 8

Speedup: 23.2×

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: 31.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

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 * ((b + sqrt(((b * b) + (c * (a * (-4.0d0)))))) / (a / (-0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 30.4%

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

   1. div-sub N/A

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

   2. flip-- N/A

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

   3. div-inv N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} - \frac{b}{a \cdot 2} \cdot \frac{b}{a \cdot 2}\right), \color{blue}{\left(\frac{1}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} + \frac{b}{a \cdot 2}}\right)}\right) \]

6. Applied egg-rr 30.9%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \frac{c}{a}\right)}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1 \cdot c}{a}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(c\right)}{a}\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(c\right)\right), a\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. neg-lowering-neg.f64 99.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(c\right), a\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.2%

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

    1. frac-times N/A

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

    2. frac-2neg N/A

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

    3. *-rgt-identity N/A

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

    4. remove-double-neg N/A

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

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(\frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)}\right) \]

    6. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\left(a \cdot \left(\frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{1}{\frac{a}{\frac{1}{2}}}\right)\right)\right)\right) \]

    10. clear-num N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{1}{\frac{1}{\frac{\frac{1}{2}}{a}}}\right)\right)\right)\right) \]

    11. un-div-inv N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\frac{1}{\frac{\frac{1}{2}}{a}}}\right)\right)\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \left(\frac{1}{\frac{\frac{1}{2}}{a}}\right)\right)\right)\right)\right) \]

11. Applied egg-rr 99.6%

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

    1. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(c, \left(\mathsf{neg}\left(\frac{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{\frac{a}{\frac{1}{2}}} \cdot a\right)\right)\right) \]

    2. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(c, \left(\left(\mathsf{neg}\left(\frac{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{\frac{a}{\frac{1}{2}}}\right)\right) \cdot \color{blue}{a}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{b + \sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)}}{\frac{a}{\frac{1}{2}}}\right)\right), \color{blue}{a}\right)\right) \]

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

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

Alternative 2: 95.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot t\_0} + \left(-0.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{a \cdot \left(t\_0 \cdot t\_0\right)} - \frac{\left(c \cdot c\right) \cdot \frac{a}{b}}{b}\right)\right) - c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    (-
     (+
      (/ (* (* a a) (* -2.0 (* c (* c c)))) (* b t_0))
      (-
       (*
        -0.25
        (/
         (* (* a a) (* (* a a) (* (* c c) (* (* c c) 20.0))))
         (* a (* t_0 t_0))))
       (/ (* (* c c) (/ a b)) b)))
     c)
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + ((-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0)))) - (((c * c) * (a / b)) / b))) - 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
    real(8) :: t_0
    t_0 = b * (b * b)
    code = (((((a * a) * ((-2.0d0) * (c * (c * c)))) / (b * t_0)) + (((-0.25d0) * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0d0)))) / (a * (t_0 * t_0)))) - (((c * c) * (a / b)) / b))) - c) / b
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + ((-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0)))) - (((c * c) * (a / b)) / b))) - c) / b;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + ((-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0)))) - (((c * c) * (a / b)) / b))) - c) / b

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(Float64(Float64(Float64(Float64(a * a) * Float64(-2.0 * Float64(c * Float64(c * c)))) / Float64(b * t_0)) + Float64(Float64(-0.25 * Float64(Float64(Float64(a * a) * Float64(Float64(a * a) * Float64(Float64(c * c) * Float64(Float64(c * c) * 20.0)))) / Float64(a * Float64(t_0 * t_0)))) - Float64(Float64(Float64(c * c) * Float64(a / b)) / b))) - c) / b)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + ((-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0)))) - (((c * c) * (a / b)) / b))) - c) / b;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(-2.0 * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 30.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 30.4%

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

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

7. Simplified 95.5%

   \[\leadsto \color{blue}{\frac{\frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}} + \left(\left(-0.25 \cdot \frac{{a}^{4} \cdot \left({c}^{4} \cdot 20\right)}{a \cdot {b}^{6}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right) - c\right)}{b}} \]

9. Applied egg-rr 95.5%

   \[\leadsto \frac{\color{blue}{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(-0.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}}{b} \]
10. Step-by-step derivation

    1. times-frac N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), 20\right)\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \left(\frac{a}{b} \cdot \frac{c \cdot c}{b}\right)\right)\right), c\right), b\right) \]

    2. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), 20\right)\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \left(\frac{\frac{a}{b} \cdot \left(c \cdot c\right)}{b}\right)\right)\right), c\right), b\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), 20\right)\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{a}{b} \cdot \left(c \cdot c\right)\right), b\right)\right)\right), c\right), b\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), 20\right)\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{a}{b}\right), \left(c \cdot c\right)\right), b\right)\right)\right), c\right), b\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), 20\right)\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(a, b\right), \left(c \cdot c\right)\right), b\right)\right)\right), c\right), b\right) \]

    6. *-lowering-*.f64 95.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), 20\right)\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(a, b\right), \mathsf{*.f64}\left(c, c\right)\right), b\right)\right)\right), c\right), b\right) \]

11. Applied egg-rr 95.5%

    \[\leadsto \frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(-0.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} - \color{blue}{\frac{\frac{a}{b} \cdot \left(c \cdot c\right)}{b}}\right)\right) - c}{b} \]

12. Final simplification 95.5%

    \[\leadsto \frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(-0.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} - \frac{\left(c \cdot c\right) \cdot \frac{a}{b}}{b}\right)\right) - c}{b} \]
13. Add Preprocessing

Alternative 3: 95.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot t\_0} + \left(-0.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{a \cdot \left(t\_0 \cdot t\_0\right)} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    (-
     (+
      (/ (* (* a a) (* -2.0 (* c (* c c)))) (* b t_0))
      (-
       (*
        -0.25
        (/
         (* (* a a) (* (* a a) (* (* c c) (* (* c c) 20.0))))
         (* a (* t_0 t_0))))
       (/ (* a (* c c)) (* b b))))
     c)
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + ((-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0)))) - ((a * (c * c)) / (b * b)))) - 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
    real(8) :: t_0
    t_0 = b * (b * b)
    code = (((((a * a) * ((-2.0d0) * (c * (c * c)))) / (b * t_0)) + (((-0.25d0) * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0d0)))) / (a * (t_0 * t_0)))) - ((a * (c * c)) / (b * b)))) - c) / b
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + ((-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + ((-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0)))) - ((a * (c * c)) / (b * b)))) - c) / b

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(Float64(Float64(Float64(Float64(a * a) * Float64(-2.0 * Float64(c * Float64(c * c)))) / Float64(b * t_0)) + Float64(Float64(-0.25 * Float64(Float64(Float64(a * a) * Float64(Float64(a * a) * Float64(Float64(c * c) * Float64(Float64(c * c) * 20.0)))) / Float64(a * Float64(t_0 * t_0)))) - Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) - c) / b)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + ((-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0)))) - ((a * (c * c)) / (b * b)))) - c) / b;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(-2.0 * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 30.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 30.4%

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

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

7. Simplified 95.5%

   \[\leadsto \color{blue}{\frac{\frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}} + \left(\left(-0.25 \cdot \frac{{a}^{4} \cdot \left({c}^{4} \cdot 20\right)}{a \cdot {b}^{6}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right) - c\right)}{b}} \]

9. Applied egg-rr 95.5%

   \[\leadsto \frac{\color{blue}{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \left(-0.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right) - c}}{b} \]
10. Add Preprocessing

Alternative 4: 95.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot t\_0} + -0.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{a \cdot \left(t\_0 \cdot t\_0\right)}\right) - \left(c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    (-
     (+
      (/ (* (* a a) (* -2.0 (* c (* c c)))) (* b t_0))
      (*
       -0.25
       (/
        (* (* a a) (* (* a a) (* (* c c) (* (* c c) 20.0))))
        (* a (* t_0 t_0)))))
     (+ c (/ (* a (* c c)) (* b b))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + (-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0))))) - (c + ((a * (c * c)) / (b * b)))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = b * (b * b)
    code = (((((a * a) * ((-2.0d0) * (c * (c * c)))) / (b * t_0)) + ((-0.25d0) * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0d0)))) / (a * (t_0 * t_0))))) - (c + ((a * (c * c)) / (b * b)))) / b
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + (-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0))))) - (c + ((a * (c * c)) / (b * b)))) / b;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + (-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0))))) - (c + ((a * (c * c)) / (b * b)))) / b

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(Float64(Float64(Float64(Float64(a * a) * Float64(-2.0 * Float64(c * Float64(c * c)))) / Float64(b * t_0)) + Float64(-0.25 * Float64(Float64(Float64(a * a) * Float64(Float64(a * a) * Float64(Float64(c * c) * Float64(Float64(c * c) * 20.0)))) / Float64(a * Float64(t_0 * t_0))))) - Float64(c + Float64(Float64(a * Float64(c * c)) / Float64(b * b)))) / b)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = (((((a * a) * (-2.0 * (c * (c * c)))) / (b * t_0)) + (-0.25 * (((a * a) * ((a * a) * ((c * c) * ((c * c) * 20.0)))) / (a * (t_0 * t_0))))) - (c + ((a * (c * c)) / (b * b)))) / b;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(-2.0 * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 30.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 30.4%

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

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

7. Simplified 95.5%

   \[\leadsto \color{blue}{\frac{\frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}} + \left(\left(-0.25 \cdot \frac{{a}^{4} \cdot \left({c}^{4} \cdot 20\right)}{a \cdot {b}^{6}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right) - c\right)}{b}} \]

9. Applied egg-rr 95.5%

   \[\leadsto \frac{\color{blue}{\left(\frac{\left(a \cdot a\right) \cdot \left(-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + -0.25 \cdot \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot 20\right)\right)\right)}{a \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right) - \left(c + \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)}}{b} \]
10. Add Preprocessing

Alternative 5: 94.0% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 30.4%

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

   1. div-sub N/A

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

   2. flip-- N/A

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

   3. div-inv N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} - \frac{b}{a \cdot 2} \cdot \frac{b}{a \cdot 2}\right), \color{blue}{\left(\frac{1}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} + \frac{b}{a \cdot 2}}\right)}\right) \]

6. Applied egg-rr 30.9%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \frac{c}{a}\right)}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1 \cdot c}{a}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(c\right)}{a}\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(c\right)\right), a\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. neg-lowering-neg.f64 99.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(c\right), a\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.2%

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

    1. frac-times N/A

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

    2. frac-2neg N/A

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

    3. *-rgt-identity N/A

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

    4. remove-double-neg N/A

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

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(\frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)}\right) \]

    6. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\left(a \cdot \left(\frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{1}{\frac{a}{\frac{1}{2}}}\right)\right)\right)\right) \]

    10. clear-num N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{1}{\frac{1}{\frac{\frac{1}{2}}{a}}}\right)\right)\right)\right) \]

    11. un-div-inv N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\frac{1}{\frac{\frac{1}{2}}{a}}}\right)\right)\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \left(\frac{1}{\frac{\frac{1}{2}}{a}}\right)\right)\right)\right)\right) \]

11. Applied egg-rr 99.6%

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

12. Taylor expanded in c around 0

    \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{\left(c \cdot \left(\frac{{a}^{2} \cdot c}{{b}^{3}} - -1 \cdot \frac{a}{b}\right) - b\right)}\right) \]
13. Step-by-step derivation

    1. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\left(c \cdot \left(\frac{{a}^{2} \cdot c}{{b}^{3}} - -1 \cdot \frac{a}{b}\right)\right), \color{blue}{b}\right)\right) \]

14. Simplified 94.5%

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

15. Final simplification 94.5%

    \[\leadsto \frac{c}{c \cdot \left(\frac{a}{b} + \frac{c}{b \cdot b} \cdot \frac{a \cdot a}{b}\right) - b} \]
16. Add Preprocessing

Alternative 6: 91.0% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 30.4%

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

   1. div-sub N/A

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

   2. flip-- N/A

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

   3. div-inv N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} \cdot \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} - \frac{b}{a \cdot 2} \cdot \frac{b}{a \cdot 2}\right), \color{blue}{\left(\frac{1}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} + \frac{b}{a \cdot 2}}\right)}\right) \]

6. Applied egg-rr 30.9%

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

7. Taylor expanded in b around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \frac{c}{a}\right)}, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1 \cdot c}{a}\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(c\right)}{a}\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(c\right)\right), a\right), \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. neg-lowering-neg.f64 99.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{neg.f64}\left(c\right), a\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, a\right), \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 99.2%

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

    1. frac-times N/A

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

    2. frac-2neg N/A

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

    3. *-rgt-identity N/A

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

    4. remove-double-neg N/A

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

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(\frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)}\right) \]

    6. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\left(a \cdot \left(\frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{\frac{1}{2}}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{\frac{1}{2}}{a}\right)\right)\right)\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{1}{\frac{a}{\frac{1}{2}}}\right)\right)\right)\right) \]

    10. clear-num N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{1}{\frac{1}{\frac{\frac{1}{2}}{a}}}\right)\right)\right)\right) \]

    11. un-div-inv N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(\frac{b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\frac{1}{\frac{\frac{1}{2}}{a}}}\right)\right)\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(c, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right), \left(\frac{1}{\frac{\frac{1}{2}}{a}}\right)\right)\right)\right)\right) \]

11. Applied egg-rr 99.6%

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

12. Taylor expanded in a around 0

    \[\leadsto \mathsf{/.f64}\left(c, \color{blue}{\left(\frac{a \cdot c}{b} - b\right)}\right) \]
13. Step-by-step derivation

    1. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\left(\frac{a \cdot c}{b}\right), \color{blue}{b}\right)\right) \]

    2. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\right), b\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\right), b\right)\right) \]

    4. *-lowering-*.f64 92.0%

       \[\leadsto \mathsf{/.f64}\left(c, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\right), b\right)\right) \]

14. Simplified 92.0%

    \[\leadsto \frac{c}{\color{blue}{\frac{c \cdot a}{b} - b}} \]
15. Add Preprocessing

Alternative 7: 81.2% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{0 - b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c (- 0.0 b)))

C

double code(double a, double b, double c) {
	return c / (0.0 - b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / (0.0d0 - b)
end function

Java

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

Python

def code(a, b, c):
	return c / (0.0 - b)

Julia

function code(a, b, c)
	return Float64(c / Float64(0.0 - b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / (0.0 - b);
end

Wolfram

code[a_, b_, c_] := N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 30.4%

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

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

7. Simplified 95.5%

   \[\leadsto \color{blue}{\frac{\frac{\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}} + \left(\left(-0.25 \cdot \frac{{a}^{4} \cdot \left({c}^{4} \cdot 20\right)}{a \cdot {b}^{6}} - \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right) - c\right)}{b}} \]

8. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot c\right)}, b\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(c\right)\right), b\right) \]

   2. neg-lowering-neg.f64 81.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(c\right), b\right) \]

10. Simplified 81.9%

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

11. Final simplification 81.9%

    \[\leadsto \frac{c}{0 - b} \]
12. Add Preprocessing

Alternative 8: 10.1% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{0 - a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b (- 0.0 a)))

C

double code(double a, double b, double c) {
	return b / (0.0 - 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 / (0.0d0 - a)
end function

Java

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

Python

def code(a, b, c):
	return b / (0.0 - a)

Julia

function code(a, b, c)
	return Float64(b / Float64(0.0 - a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / (0.0 - a);
end

Wolfram

code[a_, b_, c_] := N[(b / N[(0.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   11. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 30.4%

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

5. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{b}{a}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{b}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

   3. mul-1-neg N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(-1 \cdot a\right)}\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(b, \left(\mathsf{neg}\left(a\right)\right)\right) \]

   6. neg-lowering-neg.f64 10.1%

      \[\leadsto \mathsf{/.f64}\left(b, \mathsf{neg.f64}\left(a\right)\right) \]

7. Simplified 10.1%

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

8. Final simplification 10.1%

   \[\leadsto \frac{b}{0 - a} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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)))