Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.8%

Time: 14.9s

Alternatives: 9

Speedup: 29.0×

• Rival profile vector overflowed, profile may not be complete

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: 97.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

   1. *-commutative 18.5%

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

3. Simplified 18.5%

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

5. Taylor expanded in b around inf 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 2: 97.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a, b, c):
	return c * ((c * ((c * (math.pow(a, 3.0) * ((-5.0 * (c / math.pow(b, 7.0))) - (2.0 / (a * math.pow(b, 5.0)))))) - (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((a ^ 3.0) * Float64(Float64(-5.0 * Float64(c / (b ^ 7.0))) - Float64(2.0 / Float64(a * (b ^ 5.0)))))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)))
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(c * N[(N[(c * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[(-5.0 * N[(c / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 18.5%

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

   1. *-commutative 18.5%

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

3. Simplified 18.5%

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

5. Taylor expanded in b around inf 98.3%

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

7. Simplified 98.3%

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

8. Taylor expanded in c around 0 98.0%

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

9. Taylor expanded in a around inf 98.0%

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

    1. associate-*r/ 98.0%

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

    2. metadata-eval 98.0%

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

11. Simplified 98.0%

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

    1. associate-*r/ 98.0%

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

13. Applied egg-rr 98.0%

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

    1. mul-1-neg 98.0%

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

15. Simplified 98.0%

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

16. Final simplification 98.0%

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

Alternative 3: 97.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

   1. *-commutative 18.5%

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

3. Simplified 18.5%

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

5. Taylor expanded in a around 0 97.4%

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

6. Taylor expanded in c around inf 97.4%

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

7. Final simplification 97.4%

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

Alternative 4: 96.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

   1. *-commutative 18.5%

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

3. Simplified 18.5%

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

5. Taylor expanded in b around inf 98.3%

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

7. Simplified 98.3%

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

8. Taylor expanded in c around 0 98.0%

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

9. Taylor expanded in a around inf 98.0%

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

    1. associate-*r/ 98.0%

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

    2. metadata-eval 98.0%

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

11. Simplified 98.0%

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

12. Taylor expanded in b around inf 97.0%

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

    1. mul-1-neg 97.0%

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

    2. unsub-neg 97.0%

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

    3. associate-/l* 97.0%

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

    4. unpow2 97.0%

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

    5. unpow2 97.0%

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

    6. unpow2 97.0%

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

    7. times-frac 97.0%

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

    8. swap-sqr 97.0%

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

    9. unpow1 97.0%

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

    10. pow-plus 97.0%

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

    11. associate-*r/ 97.0%

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

    12. *-commutative 97.0%

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

    13. metadata-eval 97.0%

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

    14. *-commutative 97.0%

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

14. Simplified 97.0%

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

15. Final simplification 97.0%

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

Alternative 5: 95.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

   1. *-commutative 18.5%

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

3. Simplified 18.5%

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

5. Taylor expanded in b around inf 98.3%

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

7. Simplified 98.3%

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

8. Taylor expanded in b around inf 95.3%

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

   1. distribute-lft-out 95.3%

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

   2. associate-*r/ 95.3%

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

   3. mul-1-neg 95.3%

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

   4. distribute-neg-frac2 95.3%

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

   5. +-commutative 95.3%

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

   6. associate-/l* 95.3%

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

   7. fma-define 95.3%

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

   8. unpow2 95.3%

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

   9. unpow2 95.3%

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

   10. times-frac 95.3%

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

   11. sqr-neg 95.3%

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

   12. distribute-frac-neg 95.3%

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

   13. distribute-frac-neg 95.3%

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

   14. unpow2 95.3%

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

10. Simplified 95.3%

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

11. Final simplification 95.3%

    \[\leadsto \frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b} \]
12. Add Preprocessing

Alternative 6: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

   1. *-commutative 18.5%

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

3. Simplified 18.5%

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

5. Taylor expanded in b around inf 98.3%

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

7. Simplified 98.3%

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

8. Taylor expanded in c around 0 95.3%

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

9. Final simplification 95.3%

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

Alternative 7: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.5%

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

   1. *-commutative 18.5%

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

3. Simplified 18.5%

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

5. Taylor expanded in c around 0 94.9%

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

   1. associate-*r/ 94.9%

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

   2. neg-mul-1 94.9%

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

   3. distribute-rgt-neg-in 94.9%

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

7. Simplified 94.9%

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

8. Taylor expanded in b around inf 94.9%

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

9. Final simplification 94.9%

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

Alternative 8: 95.1% 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 18.5%

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

   1. *-commutative 18.5%

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

3. Simplified 18.5%

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

5. Taylor expanded in c around 0 94.9%

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

   1. associate-*r/ 94.9%

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

   2. neg-mul-1 94.9%

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

   3. distribute-rgt-neg-in 94.9%

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

7. Simplified 94.9%

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

8. Final simplification 94.9%

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

Alternative 9: 90.5% 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 18.5%

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

   1. *-commutative 18.5%

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

3. Simplified 18.5%

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

5. Taylor expanded in b around inf 89.9%

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

   1. associate-*r/ 89.9%

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

   2. mul-1-neg 89.9%

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

7. Simplified 89.9%

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

8. Final simplification 89.9%

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

Reproduce

herbie shell --seed 2024155 
(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)))