Quadratic roots, narrow range

Percentage Accurate: 55.7% → 90.8%

Time: 20.5s

Alternatives: 9

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: 55.7% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot c}{{b}^{7}} + 2 \cdot \frac{-1}{{b}^{5}}\right)\right) + \frac{-1}{{b}^{3}}\right)\right) - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (*
    (* c c)
    (+
     (*
      c
      (* a (+ (* -5.0 (/ (* a c) (pow b 7.0))) (* 2.0 (/ -1.0 (pow b 5.0))))))
     (/ -1.0 (pow b 3.0)))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * ((c * c) * ((c * (a * ((-5.0 * ((a * c) / pow(b, 7.0))) + (2.0 * (-1.0 / pow(b, 5.0)))))) + (-1.0 / 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 * ((c * c) * ((c * (a * (((-5.0d0) * ((a * c) / (b ** 7.0d0))) + (2.0d0 * ((-1.0d0) / (b ** 5.0d0)))))) + ((-1.0d0) / (b ** 3.0d0))))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * ((c * c) * ((c * (a * ((-5.0 * ((a * c) / Math.pow(b, 7.0))) + (2.0 * (-1.0 / Math.pow(b, 5.0)))))) + (-1.0 / Math.pow(b, 3.0))))) - (c / b);
}

Python

def code(a, b, c):
	return (a * ((c * c) * ((c * (a * ((-5.0 * ((a * c) / math.pow(b, 7.0))) + (2.0 * (-1.0 / math.pow(b, 5.0)))))) + (-1.0 / math.pow(b, 3.0))))) - (c / b)

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64(Float64(c * c) * Float64(Float64(c * Float64(a * Float64(Float64(-5.0 * Float64(Float64(a * c) / (b ^ 7.0))) + Float64(2.0 * Float64(-1.0 / (b ^ 5.0)))))) + Float64(-1.0 / (b ^ 3.0))))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((c * c) * ((c * (a * ((-5.0 * ((a * c) / (b ^ 7.0))) + (2.0 * (-1.0 / (b ^ 5.0)))))) + (-1.0 / (b ^ 3.0))))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[(c * c), $MachinePrecision] * N[(N[(c * N[(a * N[(N[(-5.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in a around 0 92.5%

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

   1. +-commutative 92.5%

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

   2. mul-1-neg 92.5%

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

   3. unsub-neg 92.5%

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

7. Simplified 92.5%

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

8. Taylor expanded in c around 0 92.5%

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

9. Taylor expanded in a around 0 92.5%

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

    1. unpow2 92.5%

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

11. Applied egg-rr 92.5%

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

12. Final simplification 92.5%

    \[\leadsto a \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot c}{{b}^{7}} + 2 \cdot \frac{-1}{{b}^{5}}\right)\right) + \frac{-1}{{b}^{3}}\right)\right) - \frac{c}{b} \]
13. Add Preprocessing

Alternative 2: 87.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (a * (pow(c, 2.0) * (((c * (a * -2.0)) / pow(b, 5.0)) + (-1.0 / 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 * ((c ** 2.0d0) * (((c * (a * (-2.0d0))) / (b ** 5.0d0)) + ((-1.0d0) / (b ** 3.0d0))))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in a around 0 92.5%

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

   1. +-commutative 92.5%

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

   2. mul-1-neg 92.5%

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

   3. unsub-neg 92.5%

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

7. Simplified 92.5%

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

8. Taylor expanded in c around 0 89.2%

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

   1. associate-*r/ 89.2%

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

   2. associate-*r* 89.2%

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

   3. *-commutative 89.2%

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

10. Simplified 89.2%

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

11. Final simplification 89.2%

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

Alternative 3: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot \left(\frac{c \cdot \left(a \cdot -4\right)}{{b}^{5}} - \frac{2}{{b}^{3}}\right)\right) + 2 \cdot \frac{-1}{b}\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (*
    a
    (+
     (* a (* c (- (/ (* c (* a -4.0)) (pow b 5.0)) (/ 2.0 (pow b 3.0)))))
     (* 2.0 (/ -1.0 b)))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (c * (a * ((a * (c * (((c * (a * -4.0)) / pow(b, 5.0)) - (2.0 / pow(b, 3.0))))) + (2.0 * (-1.0 / b))))) / (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 = (c * (a * ((a * (c * (((c * (a * (-4.0d0))) / (b ** 5.0d0)) - (2.0d0 / (b ** 3.0d0))))) + (2.0d0 * ((-1.0d0) / b))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c * (a * ((a * (c * (((c * (a * -4.0)) / Math.pow(b, 5.0)) - (2.0 / Math.pow(b, 3.0))))) + (2.0 * (-1.0 / b))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (c * (a * ((a * (c * (((c * (a * -4.0)) / math.pow(b, 5.0)) - (2.0 / math.pow(b, 3.0))))) + (2.0 * (-1.0 / b))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(a * Float64(Float64(a * Float64(c * Float64(Float64(Float64(c * Float64(a * -4.0)) / (b ^ 5.0)) - Float64(2.0 / (b ^ 3.0))))) + Float64(2.0 * Float64(-1.0 / b))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (a * ((a * (c * (((c * (a * -4.0)) / (b ^ 5.0)) - (2.0 / (b ^ 3.0))))) + (2.0 * (-1.0 / b))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(a * N[(N[(a * N[(c * N[(N[(N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in c around 0 89.0%

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

6. Taylor expanded in a around 0 88.9%

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

7. Taylor expanded in c around 0 88.9%

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

   1. associate-*r/ 88.9%

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

   2. *-commutative 88.9%

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

   3. *-commutative 88.9%

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

   4. associate-*r* 88.9%

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

   5. associate-*r/ 88.9%

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

   6. metadata-eval 88.9%

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

9. Simplified 88.9%

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

10. Final simplification 88.9%

    \[\leadsto \frac{c \cdot \left(a \cdot \left(a \cdot \left(c \cdot \left(\frac{c \cdot \left(a \cdot -4\right)}{{b}^{5}} - \frac{2}{{b}^{3}}\right)\right) + 2 \cdot \frac{-1}{b}\right)\right)}{a \cdot 2} \]
11. Add Preprocessing

Alternative 4: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{{c}^{2}}{-{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.2)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (- (* a (/ (pow c 2.0) (- (pow b 3.0)))) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.2) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * (pow(c, 2.0) / -pow(b, 3.0))) - (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) :: tmp
    if (b <= 2.2d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (a * ((c ** 2.0d0) / -(b ** 3.0d0))) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.2) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (a * (Math.pow(c, 2.0) / -Math.pow(b, 3.0))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.2:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (a * (math.pow(c, 2.0) / -math.pow(b, 3.0))) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64((c ^ 2.0) / Float64(-(b ^ 3.0)))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.2)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (a * ((c ^ 2.0) / -(b ^ 3.0))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.2], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / (-N[Power[b, 3.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2000000000000002

   1. Initial program 78.8%

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

   if 2.2000000000000002 < b

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in a around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. unsub-neg 86.8%

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

      3. mul-1-neg 86.8%

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

      4. distribute-neg-frac2 86.8%

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

      5. associate-/l* 86.8%

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

   7. Simplified 86.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{{c}^{2}}{-{b}^{3}} - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.2)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (fma a (pow (/ c b) 2.0) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.2) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(a, pow((c / b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.2], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2000000000000002

   1. Initial program 78.8%

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

   if 2.2000000000000002 < b

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in a around 0 86.8%

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

      1. mul-1-neg 86.8%

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

      2. unsub-neg 86.8%

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

      3. mul-1-neg 86.8%

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

      4. distribute-neg-frac2 86.8%

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

      5. associate-/l* 86.8%

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

   7. Simplified 86.8%

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

   8. Taylor expanded in b around inf 86.7%

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

      1. distribute-lft-out 86.7%

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

      2. mul-1-neg 86.7%

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

      3. +-commutative 86.7%

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

      4. remove-double-neg 86.7%

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

      5. neg-mul-1 86.7%

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

      6. sub-neg 86.7%

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

      7. associate-/l* 86.7%

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

      8. neg-mul-1 86.7%

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

      9. fma-neg 86.7%

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

      10. unpow2 86.7%

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

      11. unpow2 86.7%

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

      12. times-frac 86.7%

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

      13. unpow2 86.7%

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

      14. remove-double-neg 86.7%

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

   10. Simplified 86.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.2)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (* c (- (/ -1.0 b) (/ (* a c) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.2) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - ((a * c) / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.2d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c * (((-1.0d0) / b) - ((a * c) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.2) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - ((a * c) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.2:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = c * ((-1.0 / b) - ((a * c) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(a * c) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.2)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = c * ((-1.0 / b) - ((a * c) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.2], 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[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2000000000000002

   1. Initial program 78.8%

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

   if 2.2000000000000002 < b

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

   3. Simplified 51.3%

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

   5. Taylor expanded in c around 0 86.5%

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

      1. associate-*r/ 86.5%

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

      2. neg-mul-1 86.5%

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

      3. distribute-rgt-neg-in 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{a \cdot c}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((a * c) / 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) - ((a * c) / (b ** 3.0d0)))
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)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in c around 0 82.6%

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

   1. associate-*r/ 82.6%

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

   2. neg-mul-1 82.6%

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

   3. distribute-rgt-neg-in 82.6%

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

7. Simplified 82.6%

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

8. Final simplification 82.6%

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

Alternative 8: 64.1% 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 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

5. Taylor expanded in b around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. mul-1-neg 64.4%

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

7. Simplified 64.4%

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

8. Final simplification 64.4%

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

Alternative 9: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. *-commutative 56.1%

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

3. Simplified 56.1%

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

   1. add-cbrt-cube 55.2%

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

   2. pow1/3 52.5%

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

   3. pow3 52.5%

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

   4. pow2 52.5%

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

   5. pow-pow 52.6%

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

   6. metadata-eval 52.6%

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

6. Applied egg-rr 52.6%

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

   1. unpow1/3 55.5%

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

8. Simplified 55.5%

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

   1. add-cube-cbrt 55.5%

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

   2. pow3 55.5%

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

   3. neg-mul-1 55.5%

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

   4. fma-define 55.5%

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

   5. pow1/3 52.6%

      \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2}}\right)}^{3} \]

   6. pow-pow 56.1%

      \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}\right)}{a \cdot 2}}\right)}^{3} \]

   7. metadata-eval 56.1%

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

   8. *-commutative 56.1%

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

   9. *-commutative 56.1%

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

10. Applied egg-rr 56.1%

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

11. Taylor expanded in c around 0 3.2%

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

    1. rem-cube-cbrt 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

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

    4. mul0-lft 3.2%

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

    5. metadata-eval 3.2%

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

13. Simplified 3.2%

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

Reproduce

herbie shell --seed 2024112 
(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)))