Quadratic roots, narrow range

Percentage Accurate: 55.6% → 99.5%

Time: 18.5s

Alternatives: 6

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 54.1%

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

   1. div-inv N/A

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

   2. flip-- N/A

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

   3. associate-*l/ N/A

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

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

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

6. Applied egg-rr 55.7%

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

7. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 99.5%

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

9. Simplified 99.5%

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

Alternative 2: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{1}{b} + \left(-2 \cdot a\right) \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right) - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.0)
   (* (/ 0.5 a) (- (sqrt (+ (* b b) (* a (* c -4.0)))) b))
   (/
    1.0
    (-
     (* a (+ (/ 1.0 b) (* (* -2.0 a) (* (/ c (* b (* b b))) -0.5))))
     (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.0) {
		tmp = (0.5 / a) * (sqrt(((b * b) + (a * (c * -4.0)))) - b);
	} else {
		tmp = 1.0 / ((a * ((1.0 / b) + ((-2.0 * a) * ((c / (b * (b * b))) * -0.5)))) - (b / c));
	}
	return tmp;
}

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) :: tmp
    if (b <= 4.0d0) then
        tmp = (0.5d0 / a) * (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b)
    else
        tmp = 1.0d0 / ((a * ((1.0d0 / b) + (((-2.0d0) * a) * ((c / (b * (b * b))) * (-0.5d0))))) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.0) {
		tmp = (0.5 / a) * (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b);
	} else {
		tmp = 1.0 / ((a * ((1.0 / b) + ((-2.0 * a) * ((c / (b * (b * b))) * -0.5)))) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.0:
		tmp = (0.5 / a) * (math.sqrt(((b * b) + (a * (c * -4.0)))) - b)
	else:
		tmp = 1.0 / ((a * ((1.0 / b) + ((-2.0 * a) * ((c / (b * (b * b))) * -0.5)))) - (b / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.0)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b));
	else
		tmp = Float64(1.0 / Float64(Float64(a * Float64(Float64(1.0 / b) + Float64(Float64(-2.0 * a) * Float64(Float64(c / Float64(b * Float64(b * b))) * -0.5)))) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.0)
		tmp = (0.5 / a) * (sqrt(((b * b) + (a * (c * -4.0)))) - b);
	else
		tmp = 1.0 / ((a * ((1.0 / b) + ((-2.0 * a) * ((c / (b * (b * b))) * -0.5)))) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.0], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a * N[(N[(1.0 / b), $MachinePrecision] + N[(N[(-2.0 * a), $MachinePrecision] * N[(N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4

   1. Initial program 81.2%

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

      1. /-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 81.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 81.2%

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

   6. Applied egg-rr 81.2%

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

   if 4 < b

   1. Initial program 47.9%

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

      1. /-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 47.9%

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

      1. flip-+ N/A

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

      2. fmm-def N/A

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

      3. *-commutative N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

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

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

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

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

   6. Applied egg-rr 47.6%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

   8. Applied egg-rr 47.9%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

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

   11. Simplified 92.8%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \left(\frac{1}{b} + \left(a \cdot -2\right) \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right) - \frac{b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{1}{b} + \left(-2 \cdot a\right) \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right) - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a \cdot \left(\frac{1}{b} + \left(-2 \cdot a\right) \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right) - \frac{b}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (- (* a (+ (/ 1.0 b) (* (* -2.0 a) (* (/ c (* b (* b b))) -0.5)))) (/ b c))))

C

double code(double a, double b, double c) {
	return 1.0 / ((a * ((1.0 / b) + ((-2.0 * a) * ((c / (b * (b * b))) * -0.5)))) - (b / c));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / ((a * ((1.0d0 / b) + (((-2.0d0) * a) * ((c / (b * (b * b))) * (-0.5d0))))) - (b / c))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((a * ((1.0 / b) + ((-2.0 * a) * ((c / (b * (b * b))) * -0.5)))) - (b / c));
}

Python

def code(a, b, c):
	return 1.0 / ((a * ((1.0 / b) + ((-2.0 * a) * ((c / (b * (b * b))) * -0.5)))) - (b / c))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a * Float64(Float64(1.0 / b) + Float64(Float64(-2.0 * a) * Float64(Float64(c / Float64(b * Float64(b * b))) * -0.5)))) - Float64(b / c)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((a * ((1.0 / b) + ((-2.0 * a) * ((c / (b * (b * b))) * -0.5)))) - (b / c));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(a * N[(N[(1.0 / b), $MachinePrecision] + N[(N[(-2.0 * a), $MachinePrecision] * N[(N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 54.1%

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

   1. flip-+ N/A

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

   2. fmm-def N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

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

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

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

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

6. Applied egg-rr 53.8%

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

   1. associate-/r* N/A

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

   2. clear-num N/A

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

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

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

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

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

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

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

8. Applied egg-rr 54.1%

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

9. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

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

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

11. Simplified 88.6%

    \[\leadsto \frac{1}{\color{blue}{a \cdot \left(\frac{1}{b} + \left(a \cdot -2\right) \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right) - \frac{b}{c}}} \]

12. Final simplification 88.6%

    \[\leadsto \frac{1}{a \cdot \left(\frac{1}{b} + \left(-2 \cdot a\right) \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right) - \frac{b}{c}} \]
13. Add Preprocessing

Alternative 4: 82.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (- (/ a b) (/ b c))))

C

double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / ((a / b) - (b / c))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((a / b) - (b / c));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 54.1%

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

   1. flip-+ N/A

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

   2. fmm-def N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

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

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

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

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

6. Applied egg-rr 53.8%

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

   1. associate-/r* N/A

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

   2. clear-num N/A

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

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

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

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

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

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

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

8. Applied egg-rr 54.1%

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

9. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

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

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

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

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

    6. /-lowering-/.f64 83.0%

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

11. Simplified 83.0%

    \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
12. Add Preprocessing

Alternative 5: 64.3% 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 54.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 54.1%

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

   1. flip-+ N/A

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

   2. fmm-def N/A

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

   3. *-commutative N/A

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

   4. clear-num N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

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

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

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

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

6. Applied egg-rr 53.8%

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

7. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

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

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

   3. mul-1-neg N/A

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

   4. neg-lowering-neg.f64 65.5%

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

9. Simplified 65.5%

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

10. Final simplification 65.5%

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

Alternative 6: 11.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 54.1%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 11.5%

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

7. Simplified 11.5%

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

8. Final simplification 11.5%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))