Quadratic roots, medium range

Percentage Accurate: 31.6% → 95.6%

Time: 15.2s

Alternatives: 9

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: 31.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-0.25 * (a * ((pow(c, 4.0) / pow(b, 7.0)) * 20.0))))) - (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 = (a * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-0.25d0) * (a * (((c ** 4.0d0) / (b ** 7.0d0)) * 20.0d0))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

5. Taylor expanded in a around 0 96.0%

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

   1. pow1 96.0%

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

   2. distribute-rgt-out 96.0%

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

   3. div-inv 96.0%

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

   4. pow-flip 96.0%

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

   5. metadata-eval 96.0%

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

   6. metadata-eval 96.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{{\left(a \cdot \left(\left({c}^{4} \cdot {b}^{-6}\right) \cdot \color{blue}{20}\right)\right)}^{1}}{b}\right)\right) \]

7. Applied egg-rr 96.0%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{{\left(a \cdot \left(\left({c}^{4} \cdot {b}^{-6}\right) \cdot 20\right)\right)}^{1}}}{b}\right)\right) \]
8. Step-by-step derivation

   1. unpow1 96.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{a \cdot \left(\left({c}^{4} \cdot {b}^{-6}\right) \cdot 20\right)}}{b}\right)\right) \]

   2. *-commutative 96.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{\left(\left({c}^{4} \cdot {b}^{-6}\right) \cdot 20\right) \cdot a}}{b}\right)\right) \]

   3. associate-*l* 96.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot \left({b}^{-6} \cdot 20\right)\right)} \cdot a}{b}\right)\right) \]

   4. associate-*l* 96.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{{c}^{4} \cdot \left(\left({b}^{-6} \cdot 20\right) \cdot a\right)}}{b}\right)\right) \]

   5. *-commutative 96.0%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{{c}^{4} \cdot \left(\color{blue}{\left(20 \cdot {b}^{-6}\right)} \cdot a\right)}{b}\right)\right) \]

9. Simplified 96.0%

   \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{{c}^{4} \cdot \left(\left(20 \cdot {b}^{-6}\right) \cdot a\right)}}{b}\right)\right) \]

10. Taylor expanded in c around 0 96.0%

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

    1. *-commutative 96.0%

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

    2. associate-/l* 96.0%

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

    3. associate-*l* 96.0%

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

12. Simplified 96.0%

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

13. Final simplification 96.0%

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

Alternative 2: 94.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (pow(c, 3.0) * ((-2.0 * (pow(a, 2.0) / pow(b, 5.0))) - ((a / pow(b, 3.0)) / c))) - (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 ** 3.0d0) * (((-2.0d0) * ((a ** 2.0d0) / (b ** 5.0d0))) - ((a / (b ** 3.0d0)) / c))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

5. Taylor expanded in a around 0 94.7%

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

6. Taylor expanded in c around inf 94.7%

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

   1. mul-1-neg 94.7%

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

   2. unsub-neg 94.7%

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

   3. associate-/r* 94.7%

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

8. Simplified 94.7%

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

9. Final simplification 94.7%

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

Alternative 3: 93.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - a \cdot \left(\frac{c}{{b}^{3}} - \left(a \cdot -2\right) \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (-
   (/ -1.0 b)
   (* a (- (/ c (pow b 3.0)) (* (* a -2.0) (/ (pow c 2.0) (pow b 5.0))))))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * ((c / pow(b, 3.0)) - ((a * -2.0) * (pow(c, 2.0) / pow(b, 5.0))))));
}

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 * (((-1.0d0) / b) - (a * ((c / (b ** 3.0d0)) - ((a * (-2.0d0)) * ((c ** 2.0d0) / (b ** 5.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (a * ((c / Math.pow(b, 3.0)) - ((a * -2.0) * (Math.pow(c, 2.0) / Math.pow(b, 5.0))))));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - (a * ((c / math.pow(b, 3.0)) - ((a * -2.0) * (math.pow(c, 2.0) / math.pow(b, 5.0))))))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(a * Float64(Float64(c / (b ^ 3.0)) - Float64(Float64(a * -2.0) * Float64((c ^ 2.0) / (b ^ 5.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - (a * ((c / (b ^ 3.0)) - ((a * -2.0) * ((c ^ 2.0) / (b ^ 5.0))))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(a * N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(a * -2.0), $MachinePrecision] * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

5. Taylor expanded in c around 0 94.3%

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

6. Taylor expanded in a around 0 94.3%

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

   1. mul-1-neg 94.3%

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

   2. unsub-neg 94.3%

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

   3. associate-/l* 94.3%

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

   4. associate-*r* 94.3%

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

8. Simplified 94.3%

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

9. Final simplification 94.3%

   \[\leadsto c \cdot \left(\frac{-1}{b} - a \cdot \left(\frac{c}{{b}^{3}} - \left(a \cdot -2\right) \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right) \]
10. Add Preprocessing

Alternative 4: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return (-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 = (-c / b) - (a * ((c ** 2.0d0) / (b ** 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

5. Taylor expanded in a around 0 92.0%

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

   1. mul-1-neg 92.0%

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

   2. unsub-neg 92.0%

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

   3. mul-1-neg 92.0%

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

   4. distribute-neg-frac2 92.0%

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

   5. associate-/l* 92.0%

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

7. Simplified 92.0%

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

8. Final simplification 92.0%

   \[\leadsto \frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \]
9. Add Preprocessing

Alternative 5: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - \left(c \cdot a\right) \cdot {b}^{-3}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (* (* c a) (pow b -3.0)))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((c * a) * 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 = c * (((-1.0d0) / b) - ((c * a) * (b ** (-3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((c * a) * Math.pow(b, -3.0)));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - ((c * a) * math.pow(b, -3.0)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(c * a) * (b ^ -3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - ((c * a) * (b ^ -3.0)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

5. Taylor expanded in c around 0 91.7%

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

   1. associate-*r/ 91.7%

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

   2. neg-mul-1 91.7%

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

   3. distribute-rgt-neg-in 91.7%

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

7. Simplified 91.7%

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

   1. div-inv 91.7%

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

   2. *-un-lft-identity 91.7%

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

   3. prod-diff 91.7%

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

   4. pow-flip 91.7%

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

   5. metadata-eval 91.7%

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

9. Applied egg-rr 91.7%

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

    1. +-commutative 91.7%

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

    2. fma-undefine 91.7%

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

    3. *-rgt-identity 91.7%

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

    4. associate-+r+ 91.7%

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

11. Simplified 91.7%

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

12. Final simplification 91.7%

    \[\leadsto c \cdot \left(\frac{-1}{b} - \left(c \cdot a\right) \cdot {b}^{-3}\right) \]
13. Add Preprocessing

Alternative 6: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-1 - \frac{c \cdot a}{{b}^{2}}}{b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 - ((c * a) / Math.pow(b, 2.0))) / b);
}

Python

def code(a, b, c):
	return c * ((-1.0 - ((c * a) / math.pow(b, 2.0))) / b)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

5. Taylor expanded in c around 0 91.7%

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

   1. associate-*r/ 91.7%

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

   2. neg-mul-1 91.7%

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

   3. distribute-rgt-neg-in 91.7%

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

7. Simplified 91.7%

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

8. Taylor expanded in b around inf 91.8%

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

9. Final simplification 91.8%

   \[\leadsto c \cdot \frac{-1 - \frac{c \cdot a}{{b}^{2}}}{b} \]
10. Add Preprocessing

Alternative 7: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

5. Taylor expanded in c around 0 91.7%

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

   1. associate-*r/ 91.7%

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

   2. neg-mul-1 91.7%

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

   3. distribute-rgt-neg-in 91.7%

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

7. Simplified 91.7%

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

8. Taylor expanded in a around inf 91.7%

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

9. Taylor expanded in a around 0 92.0%

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

    1. fma-define 92.0%

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

    2. mul-1-neg 92.0%

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

    3. fma-neg 92.0%

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

    4. associate-*r/ 92.0%

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

    5. unpow3 92.0%

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

    6. unpow2 92.0%

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

    7. associate-/r* 92.0%

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

    8. div-sub 92.0%

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

    9. unsub-neg 92.0%

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

    10. mul-1-neg 92.0%

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

    11. distribute-lft-out 92.0%

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

    12. associate-*r/ 92.0%

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

    13. mul-1-neg 92.0%

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

    14. distribute-neg-frac2 92.0%

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

11. Simplified 92.0%

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

    1. unpow2 92.0%

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

13. Applied egg-rr 92.0%

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

14. Final simplification 92.0%

    \[\leadsto \frac{\mathsf{fma}\left(a, \frac{c}{b} \cdot \frac{c}{b}, c\right)}{-b} \]
15. Add Preprocessing

Alternative 8: 81.2% 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(c / Float64(-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 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

5. Taylor expanded in b around inf 82.3%

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

   1. associate-*r/ 82.3%

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

   2. mul-1-neg 82.3%

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

7. Simplified 82.3%

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

8. Final simplification 82.3%

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

Alternative 9: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 30.2%

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

   1. *-commutative 30.2%

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

3. Simplified 30.2%

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

5. Taylor expanded in c around 0 91.7%

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

   1. associate-*r/ 91.7%

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

   2. neg-mul-1 91.7%

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

   3. distribute-rgt-neg-in 91.7%

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

7. Simplified 91.7%

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

8. Taylor expanded in a around 0 82.1%

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

   1. expm1-log1p-u 71.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)\right)} \]

   2. expm1-undefine 31.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)} - 1} \]

10. Applied egg-rr 31.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)} - 1} \]
11. Step-by-step derivation

    1. sub-neg 31.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)} + \left(-1\right)} \]

    2. metadata-eval 31.9%

       \[\leadsto e^{\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)} + \color{blue}{-1} \]

    3. +-commutative 31.9%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)}} \]

    4. log1p-undefine 31.9%

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

    5. rem-exp-log 42.2%

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

    6. associate-*r/ 42.2%

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

    7. *-commutative 42.2%

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

    8. associate-*r/ 42.2%

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

    9. mul-1-neg 42.2%

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

    10. unsub-neg 42.2%

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

12. Simplified 42.2%

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

13. Taylor expanded in c around 0 3.2%

    \[\leadsto -1 + \color{blue}{1} \]

14. Final simplification 3.2%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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)))