Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.7%

Time: 14.8s

Alternatives: 6

Speedup: 29.0×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.7% 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: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

   1. *-commutative 17.4%

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

3. Simplified 17.4%

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

5. Taylor expanded in a around 0 98.4%

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

6. Taylor expanded in c around -inf 98.4%

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

7. Final simplification 98.4%

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

Alternative 2: 97.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

   1. *-commutative 17.4%

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

3. Simplified 17.4%

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

5. Taylor expanded in a around 0 98.4%

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

6. Taylor expanded in c around -inf 98.4%

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

   1. clear-num 98.0%

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

   2. inv-pow 98.0%

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

8. Applied egg-rr 98.0%

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

9. Taylor expanded in c around 0 98.1%

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

10. Final simplification 98.1%

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

Alternative 3: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

   1. *-commutative 17.4%

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

3. Simplified 17.4%

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

5. Taylor expanded in a around 0 98.4%

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

6. Taylor expanded in c around 0 98.0%

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

   1. fma-define 98.0%

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

   2. associate-*r/ 98.0%

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

   3. mul-1-neg 98.0%

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

   4. fma-neg 98.0%

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

   5. associate-*r* 98.0%

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

   6. *-commutative 98.0%

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

8. Simplified 98.0%

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

9. Final simplification 98.0%

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

Alternative 4: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{b} \cdot a\\ c \cdot \left(\frac{-2 \cdot \left(t\_0 \cdot t\_0\right) - c \cdot a}{{b}^{3}} + \frac{-1}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ c b) a)))
   (* c (+ (/ (- (* -2.0 (* t_0 t_0)) (* c a)) (pow b 3.0)) (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double t_0 = (c / b) * a;
	return c * ((((-2.0 * (t_0 * t_0)) - (c * a)) / pow(b, 3.0)) + (-1.0 / b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = (c / b) * a
    code = c * (((((-2.0d0) * (t_0 * t_0)) - (c * a)) / (b ** 3.0d0)) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (c / b) * a;
	return c * ((((-2.0 * (t_0 * t_0)) - (c * a)) / Math.pow(b, 3.0)) + (-1.0 / b));
}

Python

def code(a, b, c):
	t_0 = (c / b) * a
	return c * ((((-2.0 * (t_0 * t_0)) - (c * a)) / math.pow(b, 3.0)) + (-1.0 / b))

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c / b) * a)
	return Float64(c * Float64(Float64(Float64(Float64(-2.0 * Float64(t_0 * t_0)) - Float64(c * a)) / (b ^ 3.0)) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (c / b) * a;
	tmp = c * ((((-2.0 * (t_0 * t_0)) - (c * a)) / (b ^ 3.0)) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision]}, N[(c * N[(N[(N[(N[(-2.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 17.4%

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

   1. *-commutative 17.4%

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

3. Simplified 17.4%

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

5. Taylor expanded in c around 0 97.7%

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

6. Taylor expanded in b around inf 97.7%

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

   1. mul-1-neg 97.7%

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

   2. unsub-neg 97.7%

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

   3. associate-/l* 97.7%

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

   4. unpow2 97.7%

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

   5. unpow2 97.7%

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

   6. unpow2 97.7%

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

   7. times-frac 97.7%

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

   8. swap-sqr 97.7%

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

   9. unpow2 97.7%

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

   10. associate-*r/ 97.7%

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

8. Simplified 97.7%

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

   1. unpow2 97.7%

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

   2. associate-/l* 97.7%

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

   3. associate-/l* 97.7%

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

10. Applied egg-rr 97.7%

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

11. Final simplification 97.7%

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

Alternative 5: 95.4% accurate, 8.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-c - (a * ((c / b) * (c / b)))) / 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 - (a * ((c / b) * (c / b)))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

   1. *-commutative 17.4%

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

3. Simplified 17.4%

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

5. Taylor expanded in b around inf 96.5%

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

   1. mul-1-neg 96.5%

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

   2. unsub-neg 96.5%

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

   3. mul-1-neg 96.5%

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

7. Simplified 96.5%

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

   1. associate-/l* 96.5%

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

9. Applied egg-rr 96.5%

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

    1. unpow2 96.5%

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

    2. unpow2 96.5%

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

    3. times-frac 96.5%

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

    4. sqr-neg 96.5%

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

    5. distribute-frac-neg 96.5%

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

    6. distribute-frac-neg 96.5%

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

    7. unpow2 96.5%

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

    8. distribute-frac-neg 96.5%

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

    9. distribute-neg-frac2 96.5%

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

11. Simplified 96.5%

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

    1. unpow2 96.5%

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

13. Applied egg-rr 96.5%

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

14. Final simplification 96.5%

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

Alternative 6: 90.6% 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 17.4%

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

   1. *-commutative 17.4%

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

3. Simplified 17.4%

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

5. Taylor expanded in b around inf 90.8%

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

   1. associate-*r/ 90.8%

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

   2. mul-1-neg 90.8%

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

7. Simplified 90.8%

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

8. Final simplification 90.8%

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

Reproduce

herbie shell --seed 2024113 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))