Quadratic roots, narrow range

Percentage Accurate: 55.2% → 99.3%

Time: 16.2s

Alternatives: 10

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: 55.2% 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: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

   1. flip-+ 55.4%

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

10. Applied egg-rr 57.8%

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

    1. associate--r- 99.4%

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

    2. unpow2 99.4%

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

    3. unpow2 99.4%

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

    4. difference-of-squares 99.4%

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

    5. +-commutative 99.4%

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

    6. neg-mul-1 99.4%

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

    7. distribute-rgt1-in 99.4%

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

    8. metadata-eval 99.4%

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

    9. mul0-lft 99.4%

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

    10. associate-*r* 99.4%

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

    11. unpow2 99.4%

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

    12. fma-neg 99.3%

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

    13. *-commutative 99.3%

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

    14. distribute-rgt-neg-in 99.3%

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

    15. metadata-eval 99.3%

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

12. Simplified 99.3%

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

13. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

15. Simplified 99.3%

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

16. Final simplification 99.3%

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

Alternative 2: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{c \cdot a}{{b}^{3}} + \frac{1}{b}\right) - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -2.6)
   (/ 1.0 (* 2.0 (/ a (- (sqrt (fma b b (* (* c a) -4.0))) b))))
   (/ 1.0 (- (* a (+ (/ (* c a) (pow b 3.0)) (/ 1.0 b))) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -2.6) {
		tmp = 1.0 / (2.0 * (a / (sqrt(fma(b, b, ((c * a) * -4.0))) - b)));
	} else {
		tmp = 1.0 / ((a * (((c * a) / pow(b, 3.0)) + (1.0 / b))) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -2.6)
		tmp = Float64(1.0 / Float64(2.0 * Float64(a / Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -4.0))) - b))));
	else
		tmp = Float64(1.0 / Float64(Float64(a * Float64(Float64(Float64(c * a) / (b ^ 3.0)) + Float64(1.0 / b))) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.6], N[(1.0 / N[(2.0 * N[(a / N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a * N[(N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2.60000000000000009

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

   3. Simplified 86.3%

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

      1. neg-sub0 86.3%

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

      2. flip3-- 85.3%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 85.3%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 85.3%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 85.3%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 85.3%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 85.3%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 85.3%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 85.3%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 85.3%

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

   8. Simplified 85.3%

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

      1. clear-num 85.3%

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

      2. inv-pow 85.3%

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

      3. distribute-frac-neg 85.3%

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

      4. pow-div 86.4%

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

      5. metadata-eval 86.4%

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

      6. pow1 86.4%

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

      7. pow2 86.4%

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

      8. associate-*l* 86.4%

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

   10. Applied egg-rr 86.4%

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

       1. unpow-1 86.4%

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

       2. *-commutative 86.4%

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

       3. *-lft-identity 86.4%

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

       4. times-frac 86.4%

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

       5. metadata-eval 86.4%

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

       6. +-commutative 86.4%

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

       7. unsub-neg 86.4%

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

       8. unpow2 86.4%

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

       9. fma-neg 86.6%

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

       10. *-commutative 86.6%

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

       11. distribute-rgt-neg-in 86.6%

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

       12. metadata-eval 86.6%

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

   12. Simplified 86.6%

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

   if -2.60000000000000009 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.4%

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

      1. *-commutative 52.4%

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

   3. Simplified 52.4%

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

      1. neg-sub0 52.4%

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

      2. flip3-- 51.7%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 51.7%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 51.7%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 51.7%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 51.7%

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

   8. Simplified 51.7%

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

      1. clear-num 51.7%

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

      2. inv-pow 51.7%

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

      3. distribute-frac-neg 51.7%

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

      4. pow-div 52.4%

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

      5. metadata-eval 52.4%

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

      6. pow1 52.4%

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

      7. pow2 52.4%

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

      8. associate-*l* 52.4%

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

   10. Applied egg-rr 52.4%

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

       1. unpow-1 52.4%

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

       2. *-commutative 52.4%

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

       3. *-lft-identity 52.4%

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

       4. times-frac 52.4%

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

       5. metadata-eval 52.4%

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

       6. +-commutative 52.4%

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

       7. unsub-neg 52.4%

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

       8. unpow2 52.4%

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

       9. fma-neg 52.4%

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

       10. *-commutative 52.4%

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

       11. distribute-rgt-neg-in 52.4%

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

       12. metadata-eval 52.4%

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

   12. Simplified 52.4%

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

   13. Taylor expanded in a around 0 91.6%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + 0.5 \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}} \]

   14. Taylor expanded in c around 0 91.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{c \cdot a}{{b}^{3}} + \frac{1}{b}\right) - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{c \cdot a}{{b}^{3}} + \frac{1}{b}\right) - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -2.6)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ 1.0 (- (* a (+ (/ (* c a) (pow b 3.0)) (/ 1.0 b))) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -2.6) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a * (((c * a) / pow(b, 3.0)) + (1.0 / b))) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -2.6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(a * Float64(Float64(Float64(c * a) / (b ^ 3.0)) + Float64(1.0 / b))) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.6], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a * N[(N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2.60000000000000009

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. +-commutative 86.3%

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

      3. sqr-neg 86.3%

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

      4. unsub-neg 86.3%

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

      5. sqr-neg 86.3%

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

      6. fma-neg 86.5%

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

      7. distribute-lft-neg-in 86.5%

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

      8. *-commutative 86.5%

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

      9. *-commutative 86.5%

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

      10. distribute-rgt-neg-in 86.5%

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

      11. metadata-eval 86.5%

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

   3. Simplified 86.5%

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

   if -2.60000000000000009 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.4%

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

      1. *-commutative 52.4%

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

   3. Simplified 52.4%

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

      1. neg-sub0 52.4%

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

      2. flip3-- 51.7%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 51.7%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 51.7%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 51.7%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 51.7%

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

   8. Simplified 51.7%

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

      1. clear-num 51.7%

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

      2. inv-pow 51.7%

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

      3. distribute-frac-neg 51.7%

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

      4. pow-div 52.4%

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

      5. metadata-eval 52.4%

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

      6. pow1 52.4%

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

      7. pow2 52.4%

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

      8. associate-*l* 52.4%

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

   10. Applied egg-rr 52.4%

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

       1. unpow-1 52.4%

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

       2. *-commutative 52.4%

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

       3. *-lft-identity 52.4%

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

       4. times-frac 52.4%

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

       5. metadata-eval 52.4%

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

       6. +-commutative 52.4%

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

       7. unsub-neg 52.4%

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

       8. unpow2 52.4%

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

       9. fma-neg 52.4%

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

       10. *-commutative 52.4%

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

       11. distribute-rgt-neg-in 52.4%

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

       12. metadata-eval 52.4%

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

   12. Simplified 52.4%

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

   13. Taylor expanded in a around 0 91.6%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + 0.5 \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}} \]

   14. Taylor expanded in c around 0 91.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{c \cdot a}{{b}^{3}} + \frac{1}{b}\right) - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -2.6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{c \cdot a}{{b}^{3}} + \frac{1}{b}\right) - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -2.6)
     t_0
     (/ 1.0 (- (* a (+ (/ (* c a) (pow b 3.0)) (/ 1.0 b))) (/ b c))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -2.6) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((a * (((c * a) / pow(b, 3.0)) + (1.0 / b))) - (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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-2.6d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / ((a * (((c * a) / (b ** 3.0d0)) + (1.0d0 / b))) - (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -2.6) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((a * (((c * a) / Math.pow(b, 3.0)) + (1.0 / b))) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -2.6:
		tmp = t_0
	else:
		tmp = 1.0 / ((a * (((c * a) / math.pow(b, 3.0)) + (1.0 / b))) - (b / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -2.6)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(a * Float64(Float64(Float64(c * a) / (b ^ 3.0)) + Float64(1.0 / b))) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -2.6)
		tmp = t_0;
	else
		tmp = 1.0 / ((a * (((c * a) / (b ^ 3.0)) + (1.0 / b))) - (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2.6], t$95$0, N[(1.0 / N[(N[(a * N[(N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2.60000000000000009

   1. Initial program 86.3%

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

   if -2.60000000000000009 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 52.4%

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

      1. *-commutative 52.4%

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

   3. Simplified 52.4%

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

      1. neg-sub0 52.4%

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

      2. flip3-- 51.7%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 51.7%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 51.7%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 51.7%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 51.7%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 51.7%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 51.7%

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

   8. Simplified 51.7%

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

      1. clear-num 51.7%

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

      2. inv-pow 51.7%

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

      3. distribute-frac-neg 51.7%

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

      4. pow-div 52.4%

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

      5. metadata-eval 52.4%

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

      6. pow1 52.4%

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

      7. pow2 52.4%

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

      8. associate-*l* 52.4%

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

   10. Applied egg-rr 52.4%

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

       1. unpow-1 52.4%

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

       2. *-commutative 52.4%

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

       3. *-lft-identity 52.4%

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

       4. times-frac 52.4%

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

       5. metadata-eval 52.4%

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

       6. +-commutative 52.4%

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

       7. unsub-neg 52.4%

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

       8. unpow2 52.4%

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

       9. fma-neg 52.4%

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

       10. *-commutative 52.4%

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

       11. distribute-rgt-neg-in 52.4%

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

       12. metadata-eval 52.4%

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

   12. Simplified 52.4%

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

   13. Taylor expanded in a around 0 91.6%

       \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c} + a \cdot \left(-2 \cdot \left(a \cdot \left(-1 \cdot \frac{c}{{b}^{3}} + 0.5 \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{b}\right)}} \]

   14. Taylor expanded in c around 0 91.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \left(\frac{c \cdot a}{{b}^{3}} + \frac{1}{b}\right) - \frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{a}}{\left(-b\right) - \mathsf{fma}\left(-2, \frac{c \cdot a}{b}, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -0.1)
     t_0
     (/ (* 2.0 (/ (* c a) a)) (- (- b) (fma -2.0 (/ (* c a) b) b))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = (2.0 * ((c * a) / a)) / (-b - fma(-2.0, ((c * a) / b), b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(c * a) / a)) / Float64(Float64(-b) - fma(-2.0, Float64(Float64(c * a) / b), b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], t$95$0, N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[(-2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 82.6%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 50.7%

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

      1. *-commutative 50.7%

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

   3. Simplified 50.7%

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

      1. neg-sub0 50.7%

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

      2. flip3-- 50.1%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 50.1%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 50.1%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 50.1%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 50.1%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 50.1%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 50.1%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 50.1%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 50.1%

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

   8. Simplified 50.1%

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

      1. flip-+ 50.3%

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

   10. Applied egg-rr 52.7%

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

       1. associate--r- 99.4%

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

       2. unpow2 99.4%

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

       3. unpow2 99.4%

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

       4. difference-of-squares 99.4%

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

       5. +-commutative 99.4%

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

       6. neg-mul-1 99.4%

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

       7. distribute-rgt1-in 99.4%

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

       8. metadata-eval 99.4%

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

       9. mul0-lft 99.4%

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

       10. associate-*r* 99.4%

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

       11. unpow2 99.4%

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

       12. fma-neg 99.4%

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

       13. *-commutative 99.4%

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

       14. distribute-rgt-neg-in 99.4%

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

       15. metadata-eval 99.4%

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

   12. Simplified 99.4%

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

   13. Taylor expanded in a around 0 86.6%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{a \cdot c}{b}\right)}}}{a \cdot 2} \]
   14. Step-by-step derivation

       1. *-commutative 86.6%

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

   15. Simplified 86.6%

       \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}}{a \cdot 2} \]
   16. Step-by-step derivation

       1. div-inv 86.5%

          \[\leadsto \color{blue}{\frac{0 \cdot \left(\left(-b\right) - b\right) + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{1}{a \cdot 2}} \]

       2. +-commutative 86.5%

          \[\leadsto \frac{\color{blue}{\left(4 \cdot a\right) \cdot c + 0 \cdot \left(\left(-b\right) - b\right)}}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

       3. associate-*l* 86.5%

          \[\leadsto \frac{\color{blue}{4 \cdot \left(a \cdot c\right)} + 0 \cdot \left(\left(-b\right) - b\right)}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

       4. *-commutative 86.5%

          \[\leadsto \frac{4 \cdot \color{blue}{\left(c \cdot a\right)} + 0 \cdot \left(\left(-b\right) - b\right)}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

       5. associate-*r* 86.5%

          \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a} + 0 \cdot \left(\left(-b\right) - b\right)}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

       6. fma-define 86.5%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(4 \cdot c, a, 0 \cdot \left(\left(-b\right) - b\right)\right)}}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

       7. mul0-lft 86.5%

          \[\leadsto \frac{\mathsf{fma}\left(4 \cdot c, a, \color{blue}{0}\right)}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

       8. +-commutative 86.5%

          \[\leadsto \frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{\left(-b\right) - \color{blue}{\left(-2 \cdot \frac{c \cdot a}{b} + b\right)}} \cdot \frac{1}{a \cdot 2} \]

       9. fma-define 86.5%

          \[\leadsto \frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{\left(-b\right) - \color{blue}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, b\right)}} \cdot \frac{1}{a \cdot 2} \]

       10. associate-/l* 86.5%

           \[\leadsto \frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{\left(-b\right) - \mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{a}{b}}, b\right)} \cdot \frac{1}{a \cdot 2} \]

   17. Applied egg-rr 86.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{\left(-b\right) - \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)} \cdot \frac{1}{a \cdot 2}} \]
   18. Step-by-step derivation

       1. *-commutative 86.5%

          \[\leadsto \color{blue}{\frac{1}{a \cdot 2} \cdot \frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{\left(-b\right) - \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)}} \]

       2. times-frac 86.6%

          \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(4 \cdot c, a, 0\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)\right)}} \]

       3. associate-*r/ 86.6%

          \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(4 \cdot c, a, 0\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b\right)\right)}} \]

       4. *-lft-identity 86.6%

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

       5. associate-/r* 86.7%

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

       6. fma-undefine 86.7%

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

       7. +-rgt-identity 86.7%

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

       8. associate-*r* 86.7%

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

       9. *-commutative 86.7%

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

       10. *-commutative 86.7%

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

       11. times-frac 86.7%

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

       12. metadata-eval 86.7%

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

       13. *-commutative 86.7%

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

       14. associate-*r/ 86.7%

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

   19. Simplified 86.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{a}}{\left(-b\right) - \mathsf{fma}\left(-2, \frac{c \cdot a}{b}, b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -0.1) t_0 (/ 1.0 (/ (- (/ (* c a) b) b) c)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((((c * a) / b) - 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.1d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / ((((c * a) / b) - b) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.1) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.1:
		tmp = t_0
	else:
		tmp = 1.0 / ((((c * a) / b) - b) / c)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.1)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * a) / b) - b) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.1)
		tmp = t_0;
	else
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.1], t$95$0, N[(1.0 / N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.10000000000000001

   1. Initial program 82.6%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 50.7%

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

      1. *-commutative 50.7%

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

   3. Simplified 50.7%

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

      1. neg-sub0 50.7%

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

      2. flip3-- 50.1%

         \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. metadata-eval 50.1%

         \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. metadata-eval 50.1%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow2 50.1%

         \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 50.1%

      \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. sub0-neg 50.1%

         \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. +-lft-identity 50.1%

         \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. mul0-lft 50.1%

         \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. +-rgt-identity 50.1%

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

   8. Simplified 50.1%

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

      1. clear-num 50.1%

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

      2. inv-pow 50.1%

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

      3. distribute-frac-neg 50.1%

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

      4. pow-div 50.7%

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

      5. metadata-eval 50.7%

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

      6. pow1 50.7%

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

      7. pow2 50.7%

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

      8. associate-*l* 50.7%

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

   10. Applied egg-rr 50.7%

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

       1. unpow-1 50.7%

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

       2. *-commutative 50.7%

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

       3. *-lft-identity 50.7%

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

       4. times-frac 50.7%

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

       5. metadata-eval 50.7%

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

       6. +-commutative 50.7%

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

       7. unsub-neg 50.7%

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

       8. unpow2 50.7%

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

       9. fma-neg 50.7%

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

       10. *-commutative 50.7%

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

       11. distribute-rgt-neg-in 50.7%

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

       12. metadata-eval 50.7%

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

   12. Simplified 50.7%

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

   13. Taylor expanded in c around 0 86.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((((c * 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 / ((((c * a) / b) - b) / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

   1. clear-num 55.2%

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

   2. inv-pow 55.2%

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

   3. distribute-frac-neg 55.2%

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

   4. pow-div 55.8%

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

   5. metadata-eval 55.8%

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

   6. pow1 55.8%

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

   7. pow2 55.8%

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

   8. associate-*l* 55.8%

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

10. Applied egg-rr 55.8%

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

    1. unpow-1 55.8%

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

    2. *-commutative 55.8%

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

    3. *-lft-identity 55.8%

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

    4. times-frac 55.8%

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

    5. metadata-eval 55.8%

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

    6. +-commutative 55.8%

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

    7. unsub-neg 55.8%

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

    8. unpow2 55.8%

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

    9. fma-neg 55.9%

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

    10. *-commutative 55.9%

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

    11. distribute-rgt-neg-in 55.9%

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

    12. metadata-eval 55.9%

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

12. Simplified 55.9%

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

13. Taylor expanded in c around 0 82.4%

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

14. Final simplification 82.4%

    \[\leadsto \frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}} \]
15. Add Preprocessing

Alternative 8: 82.4% 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 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

   1. clear-num 55.2%

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

   2. inv-pow 55.2%

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

   3. distribute-frac-neg 55.2%

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

   4. pow-div 55.8%

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

   5. metadata-eval 55.8%

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

   6. pow1 55.8%

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

   7. pow2 55.8%

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

   8. associate-*l* 55.8%

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

10. Applied egg-rr 55.8%

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

    1. unpow-1 55.8%

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

    2. *-commutative 55.8%

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

    3. *-lft-identity 55.8%

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

    4. times-frac 55.8%

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

    5. metadata-eval 55.8%

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

    6. +-commutative 55.8%

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

    7. unsub-neg 55.8%

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

    8. unpow2 55.8%

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

    9. fma-neg 55.9%

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

    10. *-commutative 55.9%

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

    11. distribute-rgt-neg-in 55.9%

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

    12. metadata-eval 55.9%

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

12. Simplified 55.9%

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

13. Taylor expanded in a around 0 82.4%

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

14. Final simplification 82.4%

    \[\leadsto \frac{1}{\frac{a}{b} - \frac{b}{c}} \]
15. Add Preprocessing

Alternative 9: 64.6% 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 55.8%

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

   1. *-commutative 55.8%

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

   2. +-commutative 55.8%

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

   3. sqr-neg 55.8%

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

   4. unsub-neg 55.8%

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

   5. sqr-neg 55.8%

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

   6. fma-neg 55.8%

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

   7. distribute-lft-neg-in 55.8%

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

   8. *-commutative 55.8%

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

   9. *-commutative 55.8%

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

   10. distribute-rgt-neg-in 55.8%

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

   11. metadata-eval 55.8%

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

3. Simplified 55.8%

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

5. Taylor expanded in b around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. mul-1-neg 64.4%

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

7. Simplified 64.4%

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

8. Final simplification 64.4%

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

Alternative 10: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

   1. neg-sub0 55.8%

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

   2. flip3-- 55.1%

      \[\leadsto \frac{\color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. metadata-eval 55.1%

      \[\leadsto \frac{\frac{\color{blue}{0} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. metadata-eval 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow2 55.1%

      \[\leadsto \frac{\frac{0 - {b}^{3}}{0 + \left(\color{blue}{{b}^{2}} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0 - {b}^{3}}{0 + \left({b}^{2} + 0 \cdot b\right)}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. sub0-neg 55.1%

      \[\leadsto \frac{\frac{\color{blue}{-{b}^{3}}}{0 + \left({b}^{2} + 0 \cdot b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. +-lft-identity 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{\color{blue}{{b}^{2} + 0 \cdot b}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. mul0-lft 55.1%

      \[\leadsto \frac{\frac{-{b}^{3}}{{b}^{2} + \color{blue}{0}} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. +-rgt-identity 55.1%

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

8. Simplified 55.1%

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

9. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

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

    4. mul0-lft 3.2%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.2%

       \[\leadsto \frac{\color{blue}{0}}{a} \]

11. Simplified 3.2%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(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)))