Quadratic roots, medium range

Percentage Accurate: 31.5% → 99.3%

Time: 13.4s

Alternatives: 4

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.5% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -64}{\mathsf{fma}\left(4, a \cdot \left(c \cdot \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right)\right), {b}^{4}\right)}}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (* 4.0 (* a c))
   (-
    (- b)
    (sqrt
     (/
      (+ (pow b 6.0) (* (pow (* a c) 3.0) -64.0))
      (fma 4.0 (* a (* c (fma b b (* c (* 4.0 a))))) (pow b 4.0))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return ((4.0 * (a * c)) / (-b - sqrt(((pow(b, 6.0) + (pow((a * c), 3.0) * -64.0)) / fma(4.0, (a * (c * fma(b, b, (c * (4.0 * a))))), pow(b, 4.0)))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(4.0 * Float64(a * c)) / Float64(Float64(-b) - sqrt(Float64(Float64((b ^ 6.0) + Float64((Float64(a * c) ^ 3.0) * -64.0)) / fma(4.0, Float64(a * Float64(c * fma(b, b, Float64(c * Float64(4.0 * a))))), (b ^ 4.0)))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(N[Power[b, 6.0], $MachinePrecision] + N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] * -64.0), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(a * N[(c * N[(b * b + N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

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

   1. flip3-- 31.9%

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

   2. clear-num 31.9%

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

   3. pow2 31.9%

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

   4. pow2 31.9%

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

   5. pow-prod-up 32.0%

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

   6. metadata-eval 32.0%

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

   7. distribute-rgt-out 32.0%

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

   8. associate-*l* 32.0%

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

   9. +-commutative 32.0%

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

   10. fma-def 32.0%

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

   11. associate-*l* 32.0%

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

3. Applied egg-rr 31.8%

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

   1. flip-+ 31.7%

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

5. Applied egg-rr 32.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \frac{1}{{b}^{4} + 4 \cdot \left(\left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)\right)} \cdot \left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right)}{\left(-b\right) - \sqrt{\frac{1}{{b}^{4} + 4 \cdot \left(\left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)\right)} \cdot \left({b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}\right)}}}}{2 \cdot a} \]
6. Step-by-step derivation

7. Simplified 32.8%

   \[\leadsto \frac{\color{blue}{\frac{b \cdot b - \frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -64}{\mathsf{fma}\left(4, a \cdot \left(c \cdot \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right)\right), {b}^{4}\right)}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -64}{\mathsf{fma}\left(4, a \cdot \left(c \cdot \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right)\right), {b}^{4}\right)}}}}}{2 \cdot a} \]

8. Taylor expanded in b around 0 99.4%

   \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -64}{\mathsf{fma}\left(4, a \cdot \left(c \cdot \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right)\right), {b}^{4}\right)}}}}{2 \cdot a} \]

9. Final simplification 99.4%

   \[\leadsto \frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} + {\left(a \cdot c\right)}^{3} \cdot -64}{\mathsf{fma}\left(4, a \cdot \left(c \cdot \mathsf{fma}\left(b, b, c \cdot \left(4 \cdot a\right)\right)\right), {b}^{4}\right)}}}}{a \cdot 2} \]

Alternative 2: 94.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((-2.0d0) * (a * a)) / ((b ** 5.0d0) / (c ** 3.0d0))) - (c / b)) - ((a / (b ** 3.0d0)) * (c * c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

2. Taylor expanded in b around inf 94.1%

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

   1. associate-+r+ 94.1%

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

   2. mul-1-neg 94.1%

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

   3. unsub-neg 94.1%

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

   4. mul-1-neg 94.1%

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

   5. unsub-neg 94.1%

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

   6. associate-/l* 94.1%

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

   7. associate-*r/ 94.1%

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

   8. unpow2 94.1%

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

   9. associate-/l* 94.1%

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

   10. associate-/r/ 94.1%

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

   11. unpow2 94.1%

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

4. Simplified 94.1%

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

5. Final simplification 94.1%

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

Alternative 3: 90.8% accurate, 5.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * (4.0 * a)) / ((-b - b) - (-2.0 * (c * (a / 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 * (4.0d0 * a)) / ((-b - b) - ((-2.0d0) * (c * (a / b))))) / (a * 2.0d0)
end function

Java

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

Python

def code(a, b, c):
	return ((c * (4.0 * a)) / ((-b - b) - (-2.0 * (c * (a / b))))) / (a * 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

2. Taylor expanded in b around inf 21.5%

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

   1. flip-+ 21.4%

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

   2. associate-/l* 21.4%

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

   3. associate-/l* 21.4%

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

   4. associate-/l* 21.4%

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

4. Applied egg-rr 21.4%

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

   1. sqr-neg 21.4%

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

   2. associate-/r/ 21.4%

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

   3. associate-/r/ 21.4%

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

   4. associate--r+ 21.4%

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

   5. associate-/r/ 21.4%

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

6. Simplified 21.4%

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

7. Taylor expanded in b around inf 90.5%

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

   1. associate-*r* 90.5%

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

   2. *-commutative 90.5%

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

9. Simplified 90.5%

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

10. Final simplification 90.5%

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

Alternative 4: 81.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

2. Taylor expanded in b around inf 81.0%

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

   1. mul-1-neg 81.0%

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

   2. distribute-neg-frac 81.0%

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

4. Simplified 81.0%

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

5. Final simplification 81.0%

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

Reproduce

herbie shell --seed 2023279 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))