Quadratic roots, narrow range

Percentage Accurate: 55.3% → 99.5%

Time: 10.0s

Alternatives: 7

Speedup: 29.0×

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: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. flip-+ 54.3%

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

   2. pow2 54.4%

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

   3. add-sqr-sqrt 55.9%

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

   4. *-commutative 55.9%

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

   5. *-commutative 55.9%

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

   6. *-commutative 55.9%

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

   7. *-commutative 55.9%

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

3. Applied egg-rr 55.9%

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

   1. unpow2 55.9%

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

   2. sqr-neg 55.9%

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

   3. sub-neg 55.9%

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

   4. +-commutative 55.9%

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

   5. distribute-rgt-neg-in 55.9%

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

   6. fma-def 55.9%

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

   7. distribute-rgt-neg-in 55.9%

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

   8. metadata-eval 55.9%

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

   9. sub-neg 55.9%

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

   10. +-commutative 55.9%

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

   11. distribute-rgt-neg-in 55.9%

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

   12. fma-def 55.9%

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

   13. distribute-rgt-neg-in 55.9%

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

   14. metadata-eval 55.9%

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

5. Simplified 55.9%

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

6. Taylor expanded in b around 0 99.4%

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

   1. div-inv 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*r* 99.2%

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

   4. *-commutative 99.2%

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

   5. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. associate-/l* 99.5%

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

   4. *-commutative 99.5%

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

10. Simplified 99.5%

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

    1. *-un-lft-identity 99.5%

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

    2. associate-/l/ 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.1)
   (* (- (sqrt (+ (* b b) (* -4.0 (* c a)))) b) (/ 0.5 a))
   (/ (/ c (+ (* -0.5 (/ b a)) (* 0.5 (/ c b)))) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.1) {
		tmp = (sqrt(((b * b) + (-4.0 * (c * a)))) - b) * (0.5 / a);
	} else {
		tmp = (c / ((-0.5 * (b / a)) + (0.5 * (c / b)))) / (a * 2.0);
	}
	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 (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-0.1d0)) then
        tmp = (sqrt(((b * b) + ((-4.0d0) * (c * a)))) - b) * (0.5d0 / a)
    else
        tmp = (c / (((-0.5d0) * (b / a)) + (0.5d0 * (c / b)))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.1) {
		tmp = (Math.sqrt(((b * b) + (-4.0 * (c * a)))) - b) * (0.5 / a);
	} else {
		tmp = (c / ((-0.5 * (b / a)) + (0.5 * (c / b)))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.1:
		tmp = (math.sqrt(((b * b) + (-4.0 * (c * a)))) - b) * (0.5 / a)
	else:
		tmp = (c / ((-0.5 * (b / a)) + (0.5 * (c / b)))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.1)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(c / Float64(Float64(-0.5 * Float64(b / a)) + Float64(0.5 * Float64(c / b)))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.1)
		tmp = (sqrt(((b * b) + (-4.0 * (c * a)))) - b) * (0.5 / a);
	else
		tmp = (c / ((-0.5 * (b / a)) + (0.5 * (c / b)))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[(N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.10000000000000001

   1. Initial program 83.3%

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

      1. /-rgt-identity 83.3%

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

      2. metadata-eval 83.3%

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

      3. associate-/l* 83.3%

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

      4. associate-*r/ 83.3%

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

      5. +-commutative 83.3%

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

      6. unsub-neg 83.3%

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

      7. fma-neg 83.4%

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

      8. associate-*l* 83.4%

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

      9. *-commutative 83.4%

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

      10. distribute-rgt-neg-in 83.4%

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

      11. metadata-eval 83.4%

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

      12. associate-/r* 83.4%

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

      13. metadata-eval 83.4%

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

      14. metadata-eval 83.4%

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

   3. Simplified 83.4%

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

      1. fma-udef 83.3%

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

      2. *-commutative 83.3%

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

   5. Applied egg-rr 83.3%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 47.8%

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

      1. flip-+ 47.5%

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

      2. pow2 47.6%

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

      3. add-sqr-sqrt 49.1%

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

      4. *-commutative 49.1%

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

      5. *-commutative 49.1%

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

      6. *-commutative 49.1%

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

      7. *-commutative 49.1%

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

   3. Applied egg-rr 49.1%

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

      1. unpow2 49.1%

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

      2. sqr-neg 49.1%

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

      3. sub-neg 49.1%

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

      4. +-commutative 49.1%

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

      5. distribute-rgt-neg-in 49.1%

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

      6. fma-def 49.1%

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

      7. distribute-rgt-neg-in 49.1%

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

      8. metadata-eval 49.1%

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

      9. sub-neg 49.1%

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

      10. +-commutative 49.1%

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

      11. distribute-rgt-neg-in 49.1%

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

      12. fma-def 49.1%

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

      13. distribute-rgt-neg-in 49.1%

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

      14. metadata-eval 49.1%

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

   5. Simplified 49.1%

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

   6. Taylor expanded in b around 0 99.4%

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

      1. div-inv 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

      5. *-commutative 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. associate-/l* 99.5%

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

      4. *-commutative 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in b around inf 86.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}}{a \cdot 2}\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. flip-+ 54.3%

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

   2. pow2 54.4%

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

   3. add-sqr-sqrt 55.9%

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

   4. *-commutative 55.9%

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

   5. *-commutative 55.9%

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

   6. *-commutative 55.9%

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

   7. *-commutative 55.9%

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

3. Applied egg-rr 55.9%

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

   1. unpow2 55.9%

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

   2. sqr-neg 55.9%

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

   3. sub-neg 55.9%

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

   4. +-commutative 55.9%

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

   5. distribute-rgt-neg-in 55.9%

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

   6. fma-def 55.9%

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

   7. distribute-rgt-neg-in 55.9%

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

   8. metadata-eval 55.9%

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

   9. sub-neg 55.9%

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

   10. +-commutative 55.9%

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

   11. distribute-rgt-neg-in 55.9%

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

   12. fma-def 55.9%

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

   13. distribute-rgt-neg-in 55.9%

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

   14. metadata-eval 55.9%

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

5. Simplified 55.9%

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

6. Taylor expanded in b around 0 99.4%

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

   1. div-inv 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*r* 99.2%

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

   4. *-commutative 99.2%

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

   5. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. associate-/l* 99.5%

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

   4. *-commutative 99.5%

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

10. Simplified 99.5%

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

    1. associate-/r/ 99.5%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. flip-+ 54.3%

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

   2. pow2 54.4%

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

   3. add-sqr-sqrt 55.9%

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

   4. *-commutative 55.9%

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

   5. *-commutative 55.9%

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

   6. *-commutative 55.9%

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

   7. *-commutative 55.9%

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

3. Applied egg-rr 55.9%

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

   1. unpow2 55.9%

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

   2. sqr-neg 55.9%

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

   3. sub-neg 55.9%

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

   4. +-commutative 55.9%

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

   5. distribute-rgt-neg-in 55.9%

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

   6. fma-def 55.9%

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

   7. distribute-rgt-neg-in 55.9%

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

   8. metadata-eval 55.9%

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

   9. sub-neg 55.9%

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

   10. +-commutative 55.9%

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

   11. distribute-rgt-neg-in 55.9%

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

   12. fma-def 55.9%

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

   13. distribute-rgt-neg-in 55.9%

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

   14. metadata-eval 55.9%

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

5. Simplified 55.9%

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

6. Taylor expanded in b around 0 99.4%

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

   1. div-inv 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*r* 99.2%

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

   4. *-commutative 99.2%

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

   5. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. associate-/l* 99.5%

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

   4. *-commutative 99.5%

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

10. Simplified 99.5%

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

    1. fma-udef 99.5%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

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

Alternative 5: 82.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{c}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}}{a \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. flip-+ 54.3%

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

   2. pow2 54.4%

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

   3. add-sqr-sqrt 55.9%

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

   4. *-commutative 55.9%

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

   5. *-commutative 55.9%

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

   6. *-commutative 55.9%

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

   7. *-commutative 55.9%

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

3. Applied egg-rr 55.9%

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

   1. unpow2 55.9%

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

   2. sqr-neg 55.9%

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

   3. sub-neg 55.9%

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

   4. +-commutative 55.9%

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

   5. distribute-rgt-neg-in 55.9%

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

   6. fma-def 55.9%

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

   7. distribute-rgt-neg-in 55.9%

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

   8. metadata-eval 55.9%

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

   9. sub-neg 55.9%

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

   10. +-commutative 55.9%

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

   11. distribute-rgt-neg-in 55.9%

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

   12. fma-def 55.9%

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

   13. distribute-rgt-neg-in 55.9%

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

   14. metadata-eval 55.9%

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

5. Simplified 55.9%

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

6. Taylor expanded in b around 0 99.4%

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

   1. div-inv 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*r* 99.2%

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

   4. *-commutative 99.2%

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

   5. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. associate-/l* 99.5%

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

   4. *-commutative 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in b around inf 81.3%

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

12. Final simplification 81.3%

    \[\leadsto \frac{\frac{c}{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}}{a \cdot 2} \]

Alternative 6: 64.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. neg-sub0 54.6%

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

   2. associate-+l- 54.6%

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

   3. sub0-neg 54.6%

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

   4. neg-mul-1 54.6%

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

   5. associate-*l/ 54.6%

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

   6. *-commutative 54.6%

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

   7. associate-/r* 54.6%

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

   8. /-rgt-identity 54.6%

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

   9. metadata-eval 54.6%

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

3. Simplified 54.6%

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

4. Taylor expanded in b around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. neg-mul-1 64.4%

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

6. Simplified 64.4%

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

7. Final simplification 64.4%

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

Alternative 7: 1.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. neg-sub0 54.6%

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

   2. associate-+l- 54.6%

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

   3. sub0-neg 54.6%

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

   4. neg-mul-1 54.6%

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

   5. associate-*l/ 54.6%

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

   6. *-commutative 54.6%

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

   7. associate-/r* 54.6%

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

   8. /-rgt-identity 54.6%

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

   9. metadata-eval 54.6%

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

3. Simplified 54.6%

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

4. Taylor expanded in b around -inf 11.6%

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

   1. mul-1-neg 11.6%

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

   2. unsub-neg 11.6%

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

6. Simplified 11.6%

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

7. Taylor expanded in c around inf 1.6%

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

8. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2023172 
(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)))