Quadratic roots, narrow range

Percentage Accurate: 55.5% → 90.8%

Time: 12.6s

Alternatives: 7

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 4 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (*
     -0.25
     (/
      (+ (* 16.0 (* (pow a 4.0) (pow c 4.0))) (* 4.0 (pow (* a c) 4.0)))
      (* a (pow b 7.0))))
    (/ (* a (pow c 2.0)) (pow b 3.0)))
   (/ 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 * (((16.0 * (pow(a, 4.0) * pow(c, 4.0))) + (4.0 * pow((a * c), 4.0))) / (a * pow(b, 7.0)))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (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) * (((16.0d0 * ((a ** 4.0d0) * (c ** 4.0d0))) + (4.0d0 * ((a * c) ** 4.0d0))) / (a * (b ** 7.0d0)))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (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 * (((16.0 * (Math.pow(a, 4.0) * Math.pow(c, 4.0))) + (4.0 * Math.pow((a * c), 4.0))) / (a * Math.pow(b, 7.0)))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) - (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 * (((16.0 * (math.pow(a, 4.0) * math.pow(c, 4.0))) + (4.0 * math.pow((a * c), 4.0))) / (a * math.pow(b, 7.0)))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) - (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(Float64(Float64(16.0 * Float64((a ^ 4.0) * (c ^ 4.0))) + Float64(4.0 * (Float64(a * c) ^ 4.0))) / Float64(a * (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - 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 * (((16.0 * ((a ^ 4.0) * (c ^ 4.0))) + (4.0 * ((a * c) ^ 4.0))) / (a * (b ^ 7.0)))) - ((a * (c ^ 2.0)) / (b ^ 3.0))) - (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[(N[(16.0 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. *-commutative 53.4%

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

3. Simplified 53.4%

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

4. Taylor expanded in b around inf 90.7%

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

   1. *-commutative 90.7%

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

   2. unpow-prod-down 90.7%

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

   3. pow-prod-down 90.7%

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

   4. pow-pow 90.7%

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

   5. metadata-eval 90.7%

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

   6. metadata-eval 90.7%

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

6. Applied egg-rr 90.7%

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

7. Final simplification 90.7%

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

Alternative 2: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -4.0 (/ (* (pow c 3.0) (pow a 3.0)) (pow b 5.0)))
   (+
    (* -2.0 (/ (* a c) b))
    (+
     (* -2.0 (/ (* (pow a 2.0) (pow c 2.0)) (pow b 3.0)))
     (* -0.5 (/ (pow (* a c) 4.0) (/ (pow b 7.0) 20.0))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return ((-4.0 * ((pow(c, 3.0) * pow(a, 3.0)) / pow(b, 5.0))) + ((-2.0 * ((a * c) / b)) + ((-2.0 * ((pow(a, 2.0) * pow(c, 2.0)) / pow(b, 3.0))) + (-0.5 * (pow((a * c), 4.0) / (pow(b, 7.0) / 20.0)))))) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-4.0d0) * (((c ** 3.0d0) * (a ** 3.0d0)) / (b ** 5.0d0))) + (((-2.0d0) * ((a * c) / b)) + (((-2.0d0) * (((a ** 2.0d0) * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.5d0) * (((a * c) ** 4.0d0) / ((b ** 7.0d0) / 20.0d0)))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((-4.0 * ((Math.pow(c, 3.0) * Math.pow(a, 3.0)) / Math.pow(b, 5.0))) + ((-2.0 * ((a * c) / b)) + ((-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.5 * (Math.pow((a * c), 4.0) / (Math.pow(b, 7.0) / 20.0)))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return ((-4.0 * ((math.pow(c, 3.0) * math.pow(a, 3.0)) / math.pow(b, 5.0))) + ((-2.0 * ((a * c) / b)) + ((-2.0 * ((math.pow(a, 2.0) * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.5 * (math.pow((a * c), 4.0) / (math.pow(b, 7.0) / 20.0)))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-4.0 * Float64(Float64((c ^ 3.0) * (a ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-2.0 * Float64(Float64(a * c) / b)) + Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.5 * Float64((Float64(a * c) ^ 4.0) / Float64((b ^ 7.0) / 20.0)))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-4.0 * (((c ^ 3.0) * (a ^ 3.0)) / (b ^ 5.0))) + ((-2.0 * ((a * c) / b)) + ((-2.0 * (((a ^ 2.0) * (c ^ 2.0)) / (b ^ 3.0))) + (-0.5 * (((a * c) ^ 4.0) / ((b ^ 7.0) / 20.0)))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := 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 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. *-commutative 53.4%

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

3. Simplified 53.4%

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

4. Taylor expanded in b around inf 90.5%

   \[\leadsto \frac{\color{blue}{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{{b}^{7}}\right)\right)}}{a \cdot 2} \]

5. Taylor expanded in a around 0 90.5%

   \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \color{blue}{\frac{{a}^{4} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right)}{a \cdot 2} \]
6. Step-by-step derivation

   1. distribute-rgt-out 90.5%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{{a}^{4} \cdot \color{blue}{\left({c}^{4} \cdot \left(4 + 16\right)\right)}}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   2. associate-*r* 90.5%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(4 + 16\right)}}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   3. metadata-eval 90.5%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left({a}^{4} \cdot {c}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   4. pow-sqr 90.5%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left({a}^{4} \cdot \color{blue}{\left({c}^{2} \cdot {c}^{2}\right)}\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   5. metadata-eval 90.5%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left({a}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left({c}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   6. pow-sqr 90.5%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left(\color{blue}{\left({a}^{2} \cdot {a}^{2}\right)} \cdot \left({c}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   7. unswap-sqr 90.5%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)} \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   8. unpow2 90.5%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left(\left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   9. unpow2 90.5%

      \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left(\left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   10. swap-sqr 90.5%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left(\color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   11. unpow2 90.5%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   12. unpow2 90.5%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left({\left(a \cdot c\right)}^{2} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot {c}^{2}\right)\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   13. unpow2 90.5%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left({\left(a \cdot c\right)}^{2} \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(c \cdot c\right)}\right)\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   14. swap-sqr 90.5%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(a \cdot c\right)\right)}\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   15. unpow2 90.5%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{\left(a \cdot c\right)}^{2}}\right) \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   16. pow-sqr 90.5%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

   17. metadata-eval 90.5%

       \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{{\left(a \cdot c\right)}^{\color{blue}{4}} \cdot \left(4 + 16\right)}{{b}^{7}}\right)\right)}{a \cdot 2} \]

7. Simplified 90.5%

   \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}}\right)\right)}{a \cdot 2} \]

8. Final simplification 90.5%

   \[\leadsto \frac{-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \]

Alternative 3: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -200:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -200.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (- (/ (* -2.0 (* (pow a 2.0) (pow c 3.0))) (pow b 5.0)) (/ c b))
    (/ a (/ (pow b 3.0) (pow c 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -200.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (pow(a, 2.0) * pow(c, 3.0))) / pow(b, 5.0)) - (c / b)) - (a / (pow(b, 3.0) / pow(c, 2.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -200.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((a ^ 2.0) * (c ^ 3.0))) / (b ^ 5.0)) - Float64(c / b)) - Float64(a / Float64((b ^ 3.0) / (c ^ 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -200.0], 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[(N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $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 4 a) c)))) (*.f64 2 a)) < -200

   1. Initial program 91.2%

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

      1. sqr-neg 91.2%

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

      2. +-commutative 91.2%

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

      3. unsub-neg 91.2%

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

      4. sqr-neg 91.2%

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

      5. fma-neg 91.3%

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

      6. distribute-lft-neg-in 91.3%

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

      7. *-commutative 91.3%

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

      8. *-commutative 91.3%

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

      9. distribute-rgt-neg-in 91.3%

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

      10. metadata-eval 91.3%

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

      11. *-commutative 91.3%

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

   3. Simplified 91.3%

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

   if -200 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in b around inf 90.0%

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

      1. associate-+r+ 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

      4. mul-1-neg 90.0%

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

      5. unsub-neg 90.0%

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

      6. associate-*r/ 90.0%

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

      7. *-commutative 90.0%

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

      8. associate-/l* 90.0%

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

   6. Simplified 90.0%

      \[\leadsto \color{blue}{\left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -200:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \]

Alternative 4: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-2 \cdot \frac{a}{b} + 2 \cdot \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.8)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ -2.0 (+ (* -2.0 (/ a b)) (* 2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -2.0 / ((-2.0 * (a / b)) + (2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-2.0 / Float64(Float64(-2.0 * Float64(a / b)) + Float64(2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.8], 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[(-2.0 / N[(N[(-2.0 * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 80.1%

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

      1. sqr-neg 80.1%

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

      2. +-commutative 80.1%

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

      3. unsub-neg 80.1%

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

      4. sqr-neg 80.1%

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

      5. fma-neg 80.4%

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

      6. distribute-lft-neg-in 80.4%

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

      7. *-commutative 80.4%

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

      8. *-commutative 80.4%

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

      9. distribute-rgt-neg-in 80.4%

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

      10. metadata-eval 80.4%

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

      11. *-commutative 80.4%

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

   3. Simplified 80.4%

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

   if 7.79999999999999982 < b

   1. Initial program 46.9%

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

      1. *-commutative 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in b around inf 87.4%

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

      1. distribute-lft-out 87.4%

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

      2. associate-/l* 87.4%

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

      3. associate-/l* 87.4%

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

   6. Simplified 87.4%

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

      1. *-un-lft-identity 87.4%

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

      2. associate-/l* 87.3%

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

      3. associate-/r/ 87.3%

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

      4. fma-def 87.4%

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

      5. add-sqr-sqrt 87.4%

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

      6. pow2 87.4%

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

   8. Applied egg-rr 87.4%

      \[\leadsto \color{blue}{1 \cdot \frac{-2}{\frac{a \cdot 2}{\mathsf{fma}\left(\frac{a}{b}, c, {\left(\frac{a}{\frac{{b}^{1.5}}{c}}\right)}^{2}\right)}}} \]

   9. Taylor expanded in a around 0 87.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-2 \cdot \frac{a}{b} + 2 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-2 \cdot \frac{a}{b} + 2 \cdot \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.8)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ -2.0 (+ (* -2.0 (/ a b)) (* 2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -2.0 / ((-2.0 * (a / b)) + (2.0 * (b / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7.8d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-2.0d0) / (((-2.0d0) * (a / b)) + (2.0d0 * (b / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.8) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -2.0 / ((-2.0 * (a / b)) + (2.0 * (b / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.8:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = -2.0 / ((-2.0 * (a / b)) + (2.0 * (b / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-2.0 / Float64(Float64(-2.0 * Float64(a / b)) + Float64(2.0 * Float64(b / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.8)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -2.0 / ((-2.0 * (a / b)) + (2.0 * (b / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.8], 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[(-2.0 / N[(N[(-2.0 * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999982

   1. Initial program 80.1%

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

   if 7.79999999999999982 < b

   1. Initial program 46.9%

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

      1. *-commutative 46.9%

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

   3. Simplified 46.9%

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

   4. Taylor expanded in b around inf 87.4%

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

      1. distribute-lft-out 87.4%

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

      2. associate-/l* 87.4%

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

      3. associate-/l* 87.4%

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

   6. Simplified 87.4%

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

      1. *-un-lft-identity 87.4%

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

      2. associate-/l* 87.3%

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

      3. associate-/r/ 87.3%

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

      4. fma-def 87.4%

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

      5. add-sqr-sqrt 87.4%

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

      6. pow2 87.4%

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

   8. Applied egg-rr 87.4%

      \[\leadsto \color{blue}{1 \cdot \frac{-2}{\frac{a \cdot 2}{\mathsf{fma}\left(\frac{a}{b}, c, {\left(\frac{a}{\frac{{b}^{1.5}}{c}}\right)}^{2}\right)}}} \]

   9. Taylor expanded in a around 0 87.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{-2 \cdot \frac{a}{b} + 2 \cdot \frac{b}{c}}\\ \end{array} \]

Alternative 6: 82.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -2.0 / ((-2.0 * (a / b)) + (2.0 * (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 = (-2.0d0) / (((-2.0d0) * (a / b)) + (2.0d0 * (b / c)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

   1. *-commutative 53.4%

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

3. Simplified 53.4%

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

4. Taylor expanded in b around inf 82.3%

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

   1. distribute-lft-out 82.3%

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

   2. associate-/l* 82.3%

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

   3. associate-/l* 82.3%

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

6. Simplified 82.3%

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

   1. *-un-lft-identity 82.3%

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

   2. associate-/l* 82.3%

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

   3. associate-/r/ 82.3%

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

   4. fma-def 82.3%

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

   5. add-sqr-sqrt 82.3%

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

   6. pow2 82.3%

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

8. Applied egg-rr 82.3%

   \[\leadsto \color{blue}{1 \cdot \frac{-2}{\frac{a \cdot 2}{\mathsf{fma}\left(\frac{a}{b}, c, {\left(\frac{a}{\frac{{b}^{1.5}}{c}}\right)}^{2}\right)}}} \]

9. Taylor expanded in a around 0 83.0%

   \[\leadsto 1 \cdot \frac{-2}{\color{blue}{-2 \cdot \frac{a}{b} + 2 \cdot \frac{b}{c}}} \]

10. Final simplification 83.0%

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

Alternative 7: 64.3% 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 53.4%

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

   1. *-commutative 53.4%

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

3. Simplified 53.4%

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

4. Taylor expanded in b around inf 66.1%

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

   1. mul-1-neg 66.1%

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

   2. distribute-neg-frac 66.1%

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

6. Simplified 66.1%

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

7. Final simplification 66.1%

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

Reproduce

herbie shell --seed 2023321 
(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)))