Quadratic roots, medium range

Percentage Accurate: 31.2% → 95.5%

Time: 15.4s

Alternatives: 6

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: 31.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{a \cdot {b}^{6}}\right) - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right) - c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -2.0
   (/ (* (pow c 3.0) (pow a 2.0)) (pow b 4.0))
   (-
    (-
     (* -0.25 (* (pow (* c a) 4.0) (/ 20.0 (* a (pow b 6.0)))))
     (* a (* (/ c b) (/ c b))))
    c))
  b))

C

double code(double a, double b, double c) {
	return fma(-2.0, ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 4.0)), (((-0.25 * (pow((c * a), 4.0) * (20.0 / (a * pow(b, 6.0))))) - (a * ((c / b) * (c / b)))) - c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-2.0, Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 4.0)), Float64(Float64(Float64(-0.25 * Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64(a * (b ^ 6.0))))) - Float64(a * Float64(Float64(c / b) * Float64(c / b)))) - c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.6%

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

   1. *-commutative 31.6%

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

3. Simplified 31.6%

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

   1. add-cbrt-cube 31.4%

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

   2. pow1/3 32.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 32.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 32.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 32.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 32.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 32.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

7. Taylor expanded in b around inf 95.1%

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

9. Simplified 95.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \left(\frac{{c}^{4} \cdot {a}^{4}}{{b}^{6}} \cdot \frac{20}{a}\right) - a \cdot {\left(\frac{-c}{b}\right)}^{2}\right) - c\right)}{b}} \]
10. Step-by-step derivation

    1. pow1 95.1%

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

    2. frac-times 95.1%

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

    3. pow-prod-down 95.1%

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

11. Applied egg-rr 95.1%

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

    1. unpow1 95.1%

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

    2. associate-/l* 95.1%

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

    3. *-commutative 95.1%

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

    4. *-commutative 95.1%

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

13. Simplified 95.1%

    \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(\color{blue}{-0.25 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{20}{a \cdot {b}^{6}}\right)} - a \cdot {\left(\frac{-c}{b}\right)}^{2}\right) - c\right)}{b} \]
14. Step-by-step derivation

    1. unpow2 95.1%

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

    2. distribute-frac-neg 95.1%

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

    3. distribute-frac-neg 95.1%

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

15. Applied egg-rr 95.1%

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

16. Final simplification 95.1%

    \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{4}}, \left(-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{a \cdot {b}^{6}}\right) - a \cdot \left(\frac{c}{b} \cdot \frac{c}{b}\right)\right) - c\right)}{b} \]
17. Add Preprocessing

Alternative 2: 95.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (*
     a
     (+
      (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
      (* -5.0 (/ (* a (pow c 4.0)) (pow b 7.0)))))
    (/ (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))) + (-5.0 * ((a * pow(c, 4.0)) / pow(b, 7.0))))) - (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))) + ((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))))) - ((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))) + (-5.0 * ((a * Math.pow(c, 4.0)) / Math.pow(b, 7.0))))) - (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))) + (-5.0 * ((a * math.pow(c, 4.0)) / math.pow(b, 7.0))))) - (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(-5.0 * Float64(Float64(a * (c ^ 4.0)) / (b ^ 7.0))))) - 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))) + (-5.0 * ((a * (c ^ 4.0)) / (b ^ 7.0))))) - ((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[(-5.0 * N[(N[(a * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $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 31.6%

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

   1. *-commutative 31.6%

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

   2. +-commutative 31.6%

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

   3. sqr-neg 31.6%

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

   4. unsub-neg 31.6%

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

   5. sqr-neg 31.6%

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

   6. fma-neg 31.7%

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

   7. distribute-lft-neg-in 31.7%

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

   8. *-commutative 31.7%

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

   9. *-commutative 31.7%

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

   10. distribute-rgt-neg-in 31.7%

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

   11. metadata-eval 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in a around 0 95.1%

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

6. Taylor expanded in c around 0 95.1%

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

7. Final simplification 95.1%

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

Alternative 3: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.45:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -0.45) t_0 (/ (- (* a (- (pow (/ c (- b)) 2.0))) c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.45) {
		tmp = t_0;
	} else {
		tmp = ((a * -pow((c / -b), 2.0)) - c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.45d0)) then
        tmp = t_0
    else
        tmp = ((a * -((c / -b) ** 2.0d0)) - c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.45) {
		tmp = t_0;
	} else {
		tmp = ((a * -Math.pow((c / -b), 2.0)) - c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.45:
		tmp = t_0
	else:
		tmp = ((a * -math.pow((c / -b), 2.0)) - c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.45)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(a * Float64(-(Float64(c / Float64(-b)) ^ 2.0))) - c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.45)
		tmp = t_0;
	else
		tmp = ((a * -((c / -b) ^ 2.0)) - c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.45], t$95$0, N[(N[(N[(a * (-N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.450000000000000011

   1. Initial program 78.6%

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

   if -0.450000000000000011 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 24.6%

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

      1. *-commutative 24.6%

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

      2. +-commutative 24.6%

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

      3. sqr-neg 24.6%

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

      4. unsub-neg 24.6%

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

      5. sqr-neg 24.6%

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

      6. fma-neg 24.8%

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

      7. distribute-lft-neg-in 24.8%

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

      8. *-commutative 24.8%

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

      9. *-commutative 24.8%

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

      10. distribute-rgt-neg-in 24.8%

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

      11. metadata-eval 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in a around 0 93.6%

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

      1. mul-1-neg 93.6%

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

      2. unsub-neg 93.6%

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

      3. mul-1-neg 93.6%

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

      4. distribute-neg-frac2 93.6%

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

      5. associate-/l* 93.6%

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

   7. Simplified 93.6%

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

   8. Taylor expanded in b around inf 93.6%

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

   10. Simplified 93.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.45:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{\left(\frac{c}{-b}\right)}^{2}\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 \cdot \mathsf{fma}\left(a, a \cdot \frac{c \cdot 0.5}{{b}^{3}} + \frac{0.5}{b}, \frac{b}{c} \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (*
   2.0
   (fma a (+ (* a (/ (* c 0.5) (pow b 3.0))) (/ 0.5 b)) (* (/ b c) -0.5)))))

C

double code(double a, double b, double c) {
	return 1.0 / (2.0 * fma(a, ((a * ((c * 0.5) / pow(b, 3.0))) + (0.5 / b)), ((b / c) * -0.5)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(2.0 * fma(a, Float64(Float64(a * Float64(Float64(c * 0.5) / (b ^ 3.0))) + Float64(0.5 / b)), Float64(Float64(b / c) * -0.5))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(2.0 * N[(a * N[(N[(a * N[(N[(c * 0.5), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6%

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

   1. *-commutative 31.6%

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

   2. +-commutative 31.6%

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

   3. sqr-neg 31.6%

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

   4. unsub-neg 31.6%

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

   5. sqr-neg 31.6%

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

   6. fma-neg 31.7%

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

   7. distribute-lft-neg-in 31.7%

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

   8. *-commutative 31.7%

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

   9. *-commutative 31.7%

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

   10. distribute-rgt-neg-in 31.7%

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

   11. metadata-eval 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in b around inf 93.0%

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

   1. fma-define 93.0%

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

   2. cube-prod 93.0%

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

   3. distribute-lft-out 93.0%

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

   4. *-commutative 93.0%

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

7. Simplified 93.0%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

9. Applied egg-rr 92.9%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\frac{\mathsf{fma}\left(-4, {\left(a \cdot c\right)}^{3} \cdot {b}^{-4}, -2 \cdot \mathsf{fma}\left(a, c, {\left(a \cdot c\right)}^{2} \cdot {b}^{-2}\right)\right)}{b}}\right)}^{-1}} \]
10. Step-by-step derivation

    1. unpow-1 92.9%

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

    2. *-commutative 92.9%

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

    3. *-lft-identity 92.9%

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

    4. times-frac 92.9%

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

    5. metadata-eval 92.9%

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

11. Simplified 92.9%

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

12. Taylor expanded in a around 0 93.5%

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

    1. +-commutative 93.5%

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

    2. fma-define 93.5%

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

    3. mul-1-neg 93.5%

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

    4. distribute-rgt-neg-in 93.5%

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

    5. distribute-rgt-out 93.5%

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

    6. metadata-eval 93.5%

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

    7. distribute-rgt-neg-in 93.5%

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

    8. metadata-eval 93.5%

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

    9. *-commutative 93.5%

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

    10. associate-*r/ 93.5%

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

    11. associate-*r/ 93.5%

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

    12. metadata-eval 93.5%

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

    13. *-commutative 93.5%

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

14. Simplified 93.5%

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

15. Final simplification 93.5%

    \[\leadsto \frac{1}{2 \cdot \mathsf{fma}\left(a, a \cdot \frac{c \cdot 0.5}{{b}^{3}} + \frac{0.5}{b}, \frac{b}{c} \cdot -0.5\right)} \]
16. Add Preprocessing

Alternative 5: 90.9% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.6%

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

   1. *-commutative 31.6%

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

   2. +-commutative 31.6%

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

   3. sqr-neg 31.6%

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

   4. unsub-neg 31.6%

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

   5. sqr-neg 31.6%

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

   6. fma-neg 31.7%

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

   7. distribute-lft-neg-in 31.7%

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

   8. *-commutative 31.7%

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

   9. *-commutative 31.7%

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

   10. distribute-rgt-neg-in 31.7%

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

   11. metadata-eval 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in b around inf 93.0%

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

   1. fma-define 93.0%

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

   2. cube-prod 93.0%

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

   3. distribute-lft-out 93.0%

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

   4. *-commutative 93.0%

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

7. Simplified 93.0%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

9. Applied egg-rr 92.9%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\frac{\mathsf{fma}\left(-4, {\left(a \cdot c\right)}^{3} \cdot {b}^{-4}, -2 \cdot \mathsf{fma}\left(a, c, {\left(a \cdot c\right)}^{2} \cdot {b}^{-2}\right)\right)}{b}}\right)}^{-1}} \]
10. Step-by-step derivation

    1. unpow-1 92.9%

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

    2. *-commutative 92.9%

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

    3. *-lft-identity 92.9%

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

    4. times-frac 92.9%

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

    5. metadata-eval 92.9%

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

11. Simplified 92.9%

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

12. Taylor expanded in a around 0 90.0%

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

    1. +-commutative 90.0%

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

    2. mul-1-neg 90.0%

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

    3. unsub-neg 90.0%

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

14. Simplified 90.0%

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

15. Final simplification 90.0%

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

Alternative 6: 81.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(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 31.6%

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

   1. *-commutative 31.6%

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

   2. +-commutative 31.6%

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

   3. sqr-neg 31.6%

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

   4. unsub-neg 31.6%

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

   5. sqr-neg 31.6%

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

   6. fma-neg 31.7%

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

   7. distribute-lft-neg-in 31.7%

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

   8. *-commutative 31.7%

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

   9. *-commutative 31.7%

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

   10. distribute-rgt-neg-in 31.7%

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

   11. metadata-eval 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in b around inf 80.6%

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

   1. associate-*r/ 80.6%

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

   2. mul-1-neg 80.6%

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

7. Simplified 80.6%

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

8. Final simplification 80.6%

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

Reproduce

herbie shell --seed 2024141 
(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)))