Quadratic roots, medium range

Percentage Accurate: 32.0% → 95.2%

Time: 12.9s

Alternatives: 8

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 32.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}} - \frac{\frac{c}{b}}{\frac{b}{c}} \cdot \frac{a}{b}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (* -0.25 (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 20.0))))
    (* (/ (/ c b) (/ b c)) (/ a b)))
   (/ c b))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0)))) - (((c / b) / (b / c)) * (a / b))) - (c / b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * ((a ** 3.0d0) / ((b ** 7.0d0) / ((c ** 4.0d0) * 20.0d0)))) - (((c / b) / (b / c)) * (a / b))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * (Math.pow(a, 3.0) / (Math.pow(b, 7.0) / (Math.pow(c, 4.0) * 20.0)))) - (((c / b) / (b / c)) * (a / b))) - (c / b));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * (math.pow(a, 3.0) / (math.pow(b, 7.0) / (math.pow(c, 4.0) * 20.0)))) - (((c / b) / (b / c)) * (a / b))) - (c / b))

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 20.0)))) - Float64(Float64(Float64(c / b) / Float64(b / c)) * Float64(a / b))) - Float64(c / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * ((a ^ 3.0) / ((b ^ 7.0) / ((c ^ 4.0) * 20.0)))) - (((c / b) / (b / c)) * (a / b))) - (c / b));
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c / b), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in b around inf 95.9%

   \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

5. Taylor expanded in a around 0 95.9%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{a}^{3} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right) \]
6. Step-by-step derivation

   1. associate-/l* 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{4 \cdot {c}^{4} + 16 \cdot {c}^{4}}}}\right)\right) \]

   2. distribute-rgt-out 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{\color{blue}{{c}^{4} \cdot \left(4 + 16\right)}}}\right)\right) \]

   3. metadata-eval 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot \color{blue}{20}}}\right)\right) \]

7. Simplified 95.9%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}}\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{3}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   2. unpow3 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{\color{blue}{\left(b \cdot b\right) \cdot b}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   3. times-frac 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \color{blue}{\left(\frac{{c}^{2}}{b \cdot b} \cdot \frac{a}{b}\right)} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   4. unpow2 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\frac{\color{blue}{c \cdot c}}{b \cdot b} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   5. frac-times 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   6. pow1 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   7. metadata-eval 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   8. pow1 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{1}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   9. metadata-eval 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   10. pow-sqr 95.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{\left(2 \cdot \frac{2}{2}\right)}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   11. metadata-eval 95.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\left(2 \cdot \color{blue}{1}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   12. metadata-eval 95.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\color{blue}{2}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

9. Applied egg-rr 95.9%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \color{blue}{\left({\left(\frac{c}{b}\right)}^{2} \cdot \frac{a}{b}\right)} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]
10. Step-by-step derivation

    1. unpow2 95.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

    2. clear-num 95.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left(\frac{c}{b} \cdot \color{blue}{\frac{1}{\frac{b}{c}}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

    3. un-div-inv 95.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{\frac{\frac{c}{b}}{\frac{b}{c}}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

11. Applied egg-rr 95.9%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{\frac{\frac{c}{b}}{\frac{b}{c}}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

12. Final simplification 95.9%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}} - \frac{\frac{c}{b}}{\frac{b}{c}} \cdot \frac{a}{b}\right) - \frac{c}{b}\right) \]

Alternative 2: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+ (/ c b) (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) - ((c / b) + ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) - ((c / b) + ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) - ((c / b) + ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) - ((c / b) + ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - Float64(Float64(c / b) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - ((c / b) + ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in b around inf 94.3%

   \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

5. Final simplification 94.3%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

Alternative 3: 93.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c}{b}\\ \frac{-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-2 \cdot t_0 + -2 \cdot \left(t_0 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* a c) b)))
   (/
    (+
     (* -4.0 (/ (* (pow c 3.0) (pow a 3.0)) (pow b 5.0)))
     (+ (* -2.0 t_0) (* -2.0 (* t_0 (/ (* a c) (pow b 2.0))))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) / b;
	return ((-4.0 * ((pow(c, 3.0) * pow(a, 3.0)) / pow(b, 5.0))) + ((-2.0 * t_0) + (-2.0 * (t_0 * ((a * c) / pow(b, 2.0)))))) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = (a * c) / b
    code = (((-4.0d0) * (((c ** 3.0d0) * (a ** 3.0d0)) / (b ** 5.0d0))) + (((-2.0d0) * t_0) + ((-2.0d0) * (t_0 * ((a * c) / (b ** 2.0d0)))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) / b;
	return ((-4.0 * ((Math.pow(c, 3.0) * Math.pow(a, 3.0)) / Math.pow(b, 5.0))) + ((-2.0 * t_0) + (-2.0 * (t_0 * ((a * c) / Math.pow(b, 2.0)))))) / (a * 2.0);
}

Python

def code(a, b, c):
	t_0 = (a * c) / b
	return ((-4.0 * ((math.pow(c, 3.0) * math.pow(a, 3.0)) / math.pow(b, 5.0))) + ((-2.0 * t_0) + (-2.0 * (t_0 * ((a * c) / math.pow(b, 2.0)))))) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) / b)
	return Float64(Float64(Float64(-4.0 * Float64(Float64((c ^ 3.0) * (a ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-2.0 * t_0) + Float64(-2.0 * Float64(t_0 * Float64(Float64(a * c) / (b ^ 2.0)))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (a * c) / b;
	tmp = ((-4.0 * (((c ^ 3.0) * (a ^ 3.0)) / (b ^ 5.0))) + ((-2.0 * t_0) + (-2.0 * (t_0 * ((a * c) / (b ^ 2.0)))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]}, N[(N[(N[(-4.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(-2.0 * N[(t$95$0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in b around inf 93.9%

   \[\leadsto \frac{\color{blue}{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{a \cdot 2} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 93.9%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{\color{blue}{\sqrt{{a}^{2} \cdot {c}^{2}} \cdot \sqrt{{a}^{2} \cdot {c}^{2}}}}{{b}^{3}}\right)}{a \cdot 2} \]

   2. unpow3 93.9%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}} \cdot \sqrt{{a}^{2} \cdot {c}^{2}}}{\color{blue}{\left(b \cdot b\right) \cdot b}}\right)}{a \cdot 2} \]

   3. times-frac 93.9%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \color{blue}{\left(\frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)}\right)}{a \cdot 2} \]

   4. sqrt-prod 93.9%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{\color{blue}{\sqrt{{a}^{2}} \cdot \sqrt{{c}^{2}}}}{b \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   5. unpow2 93.9%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{\sqrt{\color{blue}{a \cdot a}} \cdot \sqrt{{c}^{2}}}{b \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   6. sqrt-prod 93.9%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{\color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)} \cdot \sqrt{{c}^{2}}}{b \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   7. add-sqr-sqrt 93.9%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{\color{blue}{a} \cdot \sqrt{{c}^{2}}}{b \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   8. unpow2 93.9%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot \sqrt{\color{blue}{c \cdot c}}}{b \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   9. sqrt-prod 93.9%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot \color{blue}{\left(\sqrt{c} \cdot \sqrt{c}\right)}}{b \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   10. add-sqr-sqrt 93.9%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot \color{blue}{c}}{b \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   11. pow1 93.9%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot c}{\color{blue}{{b}^{1}} \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   12. metadata-eval 93.9%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot c}{{b}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot b} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   13. pow1 93.9%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot c}{{b}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{b}^{1}}} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   14. metadata-eval 93.9%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot c}{{b}^{\left(\frac{2}{2}\right)} \cdot {b}^{\color{blue}{\left(\frac{2}{2}\right)}}} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   15. pow-sqr 93.9%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot c}{\color{blue}{{b}^{\left(2 \cdot \frac{2}{2}\right)}}} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   16. metadata-eval 93.9%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot c}{{b}^{\left(2 \cdot \color{blue}{1}\right)}} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

   17. metadata-eval 93.9%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot c}{{b}^{\color{blue}{2}}} \cdot \frac{\sqrt{{a}^{2} \cdot {c}^{2}}}{b}\right)\right)}{a \cdot 2} \]

6. Applied egg-rr 93.9%

   \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \color{blue}{\left(\frac{a \cdot c}{{b}^{2}} \cdot \frac{a \cdot c}{b}\right)}\right)}{a \cdot 2} \]

7. Final simplification 93.9%

   \[\leadsto \frac{-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \left(\frac{a \cdot c}{b} \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}{a \cdot 2} \]

Alternative 4: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;b \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{2}}{b + \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= b 4e-5)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 2.0))
     (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (b <= 4e-5) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (b <= 4e-5)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4e-5], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000033e-5

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

   3. Simplified 83.4%

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

      1. add-cbrt-cube 82.4%

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

      2. pow3 82.4%

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

      3. fma-neg 82.6%

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

      4. distribute-lft-neg-in 82.6%

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

      5. *-commutative 82.6%

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

      6. distribute-lft-neg-in 82.6%

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

      7. metadata-eval 82.6%

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

      8. associate-*r* 82.6%

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

   5. Applied egg-rr 82.6%

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

      1. +-commutative 82.6%

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

      2. flip-+ 82.4%

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

      3. add-sqr-sqrt 82.6%

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

      4. rem-cbrt-cube 84.1%

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

      5. *-commutative 84.1%

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

      6. sqr-neg 84.1%

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

      7. pow1 84.1%

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

      8. metadata-eval 84.1%

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

      9. pow1 84.1%

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

      10. metadata-eval 84.1%

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

      11. pow-sqr 84.1%

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

      12. metadata-eval 84.1%

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

      13. metadata-eval 84.1%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 1.6%

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

      16. sqr-neg 1.6%

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

   7. Applied egg-rr 84.0%

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

   if 4.00000000000000033e-5 < b

   1. Initial program 27.9%

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

      1. *-commutative 27.9%

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

   3. Simplified 27.9%

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

   4. Taylor expanded in b around inf 93.8%

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

      1. distribute-lft-out 93.8%

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

   6. Simplified 93.8%

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

      1. *-commutative 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{3}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      2. unpow3 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{\color{blue}{\left(b \cdot b\right) \cdot b}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      3. times-frac 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \color{blue}{\left(\frac{{c}^{2}}{b \cdot b} \cdot \frac{a}{b}\right)} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      4. unpow2 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\frac{\color{blue}{c \cdot c}}{b \cdot b} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      5. frac-times 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      6. pow1 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      7. metadata-eval 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      8. pow1 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{1}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      9. metadata-eval 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      10. pow-sqr 97.2%

          \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{\left(2 \cdot \frac{2}{2}\right)}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      11. metadata-eval 97.2%

          \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\left(2 \cdot \color{blue}{1}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      12. metadata-eval 97.2%

          \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\color{blue}{2}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   8. Applied egg-rr 93.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 5: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4e-5)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-5) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4e-5], 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[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000033e-5

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

   3. Simplified 83.5%

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

   if 4.00000000000000033e-5 < b

   1. Initial program 27.9%

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

      1. *-commutative 27.9%

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

   3. Simplified 27.9%

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

   4. Taylor expanded in b around inf 93.8%

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

      1. distribute-lft-out 93.8%

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

   6. Simplified 93.8%

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

      1. *-commutative 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{3}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      2. unpow3 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{\color{blue}{\left(b \cdot b\right) \cdot b}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      3. times-frac 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \color{blue}{\left(\frac{{c}^{2}}{b \cdot b} \cdot \frac{a}{b}\right)} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      4. unpow2 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\frac{\color{blue}{c \cdot c}}{b \cdot b} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      5. frac-times 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      6. pow1 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      7. metadata-eval 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      8. pow1 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{1}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      9. metadata-eval 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      10. pow-sqr 97.2%

          \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{\left(2 \cdot \frac{2}{2}\right)}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      11. metadata-eval 97.2%

          \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\left(2 \cdot \color{blue}{1}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      12. metadata-eval 97.2%

          \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\color{blue}{2}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   8. Applied egg-rr 93.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 6: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4e-5)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-5) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4d-5) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a / b) * ((c / b) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4e-5) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * Math.pow((c / b), 2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4e-5:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a / b) * math.pow((c / b), 2.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4e-5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4e-5)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((a / b) * ((c / b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4e-5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000033e-5

   1. Initial program 83.4%

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

   if 4.00000000000000033e-5 < b

   1. Initial program 27.9%

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

      1. *-commutative 27.9%

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

   3. Simplified 27.9%

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

   4. Taylor expanded in b around inf 93.8%

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

      1. distribute-lft-out 93.8%

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

   6. Simplified 93.8%

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

      1. *-commutative 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{3}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      2. unpow3 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{\color{blue}{\left(b \cdot b\right) \cdot b}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      3. times-frac 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \color{blue}{\left(\frac{{c}^{2}}{b \cdot b} \cdot \frac{a}{b}\right)} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      4. unpow2 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\frac{\color{blue}{c \cdot c}}{b \cdot b} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      5. frac-times 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      6. pow1 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      7. metadata-eval 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      8. pow1 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{1}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      9. metadata-eval 97.2%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      10. pow-sqr 97.2%

          \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{\left(2 \cdot \frac{2}{2}\right)}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      11. metadata-eval 97.2%

          \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\left(2 \cdot \color{blue}{1}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

      12. metadata-eval 97.2%

          \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\color{blue}{2}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   8. Applied egg-rr 93.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 7: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in b around inf 91.5%

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

   1. distribute-lft-out 91.5%

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

6. Simplified 91.5%

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

   1. *-commutative 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{\color{blue}{{c}^{2} \cdot a}}{{b}^{3}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   2. unpow3 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{\color{blue}{\left(b \cdot b\right) \cdot b}} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   3. times-frac 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \color{blue}{\left(\frac{{c}^{2}}{b \cdot b} \cdot \frac{a}{b}\right)} + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   4. unpow2 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\frac{\color{blue}{c \cdot c}}{b \cdot b} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   5. frac-times 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{\left(\frac{c}{b} \cdot \frac{c}{b}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   6. pow1 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left(\color{blue}{{\left(\frac{c}{b}\right)}^{1}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   7. metadata-eval 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \frac{c}{b}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   8. pow1 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left(\frac{c}{b}\right)}^{1}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   9. metadata-eval 95.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\left({\left(\frac{c}{b}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   10. pow-sqr 95.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left(\color{blue}{{\left(\frac{c}{b}\right)}^{\left(2 \cdot \frac{2}{2}\right)}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   11. metadata-eval 95.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\left(2 \cdot \color{blue}{1}\right)} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

   12. metadata-eval 95.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \left({\left(\frac{c}{b}\right)}^{\color{blue}{2}} \cdot \frac{a}{b}\right) + -0.25 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}\right)\right) \]

8. Applied egg-rr 91.5%

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

9. Final simplification 91.5%

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

Alternative 8: 80.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.5%

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

   1. *-commutative 31.5%

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

3. Simplified 31.5%

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

4. Taylor expanded in b around inf 81.5%

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

   1. mul-1-neg 81.5%

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

6. Simplified 81.5%

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

7. Final simplification 81.5%

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

Reproduce

herbie shell --seed 2023301 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))