Quadratic roots, medium range

Percentage Accurate: 30.9% → 95.7%

Time: 14.0s

Alternatives: 6

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 30.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (fma
   -0.25
   (* (/ (pow (* c a) 4.0) a) (/ 20.0 (pow b 7.0)))
   (- (/ -2.0 (/ (pow b 5.0) (* a (* a (pow c 3.0))))) (/ c b)))
  (/ (* c (* c a)) (pow b 3.0))))

C

double code(double a, double b, double c) {
	return fma(-0.25, ((pow((c * a), 4.0) / a) * (20.0 / pow(b, 7.0))), ((-2.0 / (pow(b, 5.0) / (a * (a * pow(c, 3.0))))) - (c / b))) - ((c * (c * a)) / pow(b, 3.0));
}

Julia

function code(a, b, c)
	return Float64(fma(-0.25, Float64(Float64((Float64(c * a) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0))), Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64(a * Float64(a * (c ^ 3.0))))) - Float64(c / b))) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.25 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.1%

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

   1. neg-sub0 27.1%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 27.1%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 27.1%

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

   4. neg-mul-1 27.1%

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

   5. associate-*l/ 27.1%

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

   6. *-commutative 27.1%

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

   7. associate-/r* 27.1%

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

   8. /-rgt-identity 27.1%

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

   9. metadata-eval 27.1%

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

3. Simplified 27.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 97.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

6. Simplified 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{\mathsf{fma}\left(16, {c}^{4} \cdot {a}^{4}, 4 \cdot \left({c}^{4} \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}} \]

7. Taylor expanded in b around 0 97.6%

   \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]
8. Step-by-step derivation

   1. distribute-rgt-out 97.6%

      \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

   2. times-frac 97.6%

      \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{c}^{4} \cdot {a}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

9. Simplified 97.6%

   \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

10. Final simplification 97.6%

    \[\leadsto \mathsf{fma}\left(-0.25, \frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}, \frac{-2}{\frac{{b}^{5}}{a \cdot \left(a \cdot {c}^{3}\right)}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}} \]

Alternative 2: 94.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.1%

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

   1. neg-sub0 27.1%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 27.1%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 27.1%

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

   4. neg-mul-1 27.1%

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

   5. associate-*l/ 27.1%

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

   6. *-commutative 27.1%

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

   7. associate-/r* 27.1%

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

   8. /-rgt-identity 27.1%

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

   9. metadata-eval 27.1%

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

3. Simplified 27.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 96.2%

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

   1. +-commutative 96.2%

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

   2. mul-1-neg 96.2%

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

   3. unsub-neg 96.2%

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

   4. +-commutative 96.2%

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

   5. mul-1-neg 96.2%

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

   6. unsub-neg 96.2%

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

   7. associate-*r/ 96.2%

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

   8. associate-/l* 96.2%

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

   9. *-commutative 96.2%

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

   10. unpow2 96.2%

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

   11. associate-*l* 96.2%

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

6. Simplified 96.2%

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

7. Final simplification 96.2%

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

Alternative 3: 94.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((((pow((c * a), 2.0) / b) * 6.0) + (-6.0 * ((c * a) * b))) / ((3.0 * (b * b)) - (c * (a * 6.0)))) / (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) ** 2.0d0) / b) * 6.0d0) + ((-6.0d0) * ((c * a) * b))) / ((3.0d0 * (b * b)) - (c * (a * 6.0d0)))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((((Math.pow((c * a), 2.0) / b) * 6.0) + (-6.0 * ((c * a) * b))) / ((3.0 * (b * b)) - (c * (a * 6.0)))) / (a * 2.0);
}

Python

def code(a, b, c):
	return ((((math.pow((c * a), 2.0) / b) * 6.0) + (-6.0 * ((c * a) * b))) / ((3.0 * (b * b)) - (c * (a * 6.0)))) / (a * 2.0)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((((((c * a) ^ 2.0) / b) * 6.0) + (-6.0 * ((c * a) * b))) / ((3.0 * (b * b)) - (c * (a * 6.0)))) / (a * 2.0);
end

Wolfram

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

Derivation

1. Initial program 27.1%

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

   1. flip3-+ 27.0%

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

   2. cube-neg 27.0%

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

   3. pow1/2 27.0%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\color{blue}{\left({\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{0.5}\right)}}^{3}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   4. pow-pow 28.6%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + \color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{\left(0.5 \cdot 3\right)}}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   5. *-commutative 28.6%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}\right)}^{\left(0.5 \cdot 3\right)}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   6. *-commutative 28.6%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}\right)}^{\left(0.5 \cdot 3\right)}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   7. metadata-eval 28.6%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{\color{blue}{1.5}}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   8. pow2 28.6%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{{\left(-b\right)}^{2}} + \left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

3. Applied egg-rr 28.6%

   \[\leadsto \frac{\color{blue}{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - \left(-b\right) \cdot \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}}}{2 \cdot a} \]

4. Taylor expanded in c around 0 24.8%

   \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{\left(2 \cdot {b}^{2} + -1 \cdot \left(c \cdot \left(4 \cdot a + 2 \cdot a\right)\right)\right) - -1 \cdot {b}^{2}}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. cancel-sign-sub-inv 24.8%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{\left(2 \cdot {b}^{2} + -1 \cdot \left(c \cdot \left(4 \cdot a + 2 \cdot a\right)\right)\right) + \left(--1\right) \cdot {b}^{2}}}}{2 \cdot a} \]

   2. +-commutative 24.8%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{\left(-1 \cdot \left(c \cdot \left(4 \cdot a + 2 \cdot a\right)\right) + 2 \cdot {b}^{2}\right)} + \left(--1\right) \cdot {b}^{2}}}{2 \cdot a} \]

   3. metadata-eval 24.8%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\left(-1 \cdot \left(c \cdot \left(4 \cdot a + 2 \cdot a\right)\right) + 2 \cdot {b}^{2}\right) + \color{blue}{1} \cdot {b}^{2}}}{2 \cdot a} \]

   4. *-lft-identity 24.8%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\left(-1 \cdot \left(c \cdot \left(4 \cdot a + 2 \cdot a\right)\right) + 2 \cdot {b}^{2}\right) + \color{blue}{{b}^{2}}}}{2 \cdot a} \]

   5. associate-+l+ 24.8%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{-1 \cdot \left(c \cdot \left(4 \cdot a + 2 \cdot a\right)\right) + \left(2 \cdot {b}^{2} + {b}^{2}\right)}}}{2 \cdot a} \]

   6. mul-1-neg 24.8%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{\left(-c \cdot \left(4 \cdot a + 2 \cdot a\right)\right)} + \left(2 \cdot {b}^{2} + {b}^{2}\right)}}{2 \cdot a} \]

   7. distribute-rgt-neg-in 24.8%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{c \cdot \left(-\left(4 \cdot a + 2 \cdot a\right)\right)} + \left(2 \cdot {b}^{2} + {b}^{2}\right)}}{2 \cdot a} \]

   8. distribute-rgt-out 24.8%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{c \cdot \left(-\color{blue}{a \cdot \left(4 + 2\right)}\right) + \left(2 \cdot {b}^{2} + {b}^{2}\right)}}{2 \cdot a} \]

   9. metadata-eval 24.8%

      \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{c \cdot \left(-a \cdot \color{blue}{6}\right) + \left(2 \cdot {b}^{2} + {b}^{2}\right)}}{2 \cdot a} \]

   10. *-lft-identity 24.8%

       \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{c \cdot \left(-a \cdot 6\right) + \left(2 \cdot {b}^{2} + \color{blue}{1 \cdot {b}^{2}}\right)}}{2 \cdot a} \]

   11. distribute-rgt-out 24.8%

       \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{c \cdot \left(-a \cdot 6\right) + \color{blue}{{b}^{2} \cdot \left(2 + 1\right)}}}{2 \cdot a} \]

   12. metadata-eval 24.8%

       \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{c \cdot \left(-a \cdot 6\right) + {b}^{2} \cdot \color{blue}{3}}}{2 \cdot a} \]

   13. *-commutative 24.8%

       \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{c \cdot \left(-a \cdot 6\right) + \color{blue}{3 \cdot {b}^{2}}}}{2 \cdot a} \]

   14. unpow2 24.8%

       \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \color{blue}{\left(b \cdot b\right)}}}{2 \cdot a} \]

6. Simplified 24.8%

   \[\leadsto \frac{\frac{\left(-{b}^{3}\right) + {\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{1.5}}{\color{blue}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}}{2 \cdot a} \]

7. Taylor expanded in b around inf 25.2%

   \[\leadsto \frac{\frac{\color{blue}{18 \cdot \frac{{c}^{2} \cdot {a}^{2}}{b} + \left(-12 \cdot \frac{{c}^{2} \cdot {a}^{2}}{b} + \left({b}^{3} + \left(-6 \cdot \left(c \cdot \left(a \cdot b\right)\right) + -1 \cdot {b}^{3}\right)\right)\right)}}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]
8. Step-by-step derivation

   1. associate-+r+ 25.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(18 \cdot \frac{{c}^{2} \cdot {a}^{2}}{b} + -12 \cdot \frac{{c}^{2} \cdot {a}^{2}}{b}\right) + \left({b}^{3} + \left(-6 \cdot \left(c \cdot \left(a \cdot b\right)\right) + -1 \cdot {b}^{3}\right)\right)}}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]

   2. distribute-rgt-out 25.2%

      \[\leadsto \frac{\frac{\color{blue}{\frac{{c}^{2} \cdot {a}^{2}}{b} \cdot \left(18 + -12\right)} + \left({b}^{3} + \left(-6 \cdot \left(c \cdot \left(a \cdot b\right)\right) + -1 \cdot {b}^{3}\right)\right)}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]

   3. unpow2 25.2%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}}{b} \cdot \left(18 + -12\right) + \left({b}^{3} + \left(-6 \cdot \left(c \cdot \left(a \cdot b\right)\right) + -1 \cdot {b}^{3}\right)\right)}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]

   4. unpow2 25.2%

      \[\leadsto \frac{\frac{\frac{\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}}{b} \cdot \left(18 + -12\right) + \left({b}^{3} + \left(-6 \cdot \left(c \cdot \left(a \cdot b\right)\right) + -1 \cdot {b}^{3}\right)\right)}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]

   5. unswap-sqr 25.2%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}{b} \cdot \left(18 + -12\right) + \left({b}^{3} + \left(-6 \cdot \left(c \cdot \left(a \cdot b\right)\right) + -1 \cdot {b}^{3}\right)\right)}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]

   6. unpow2 25.2%

      \[\leadsto \frac{\frac{\frac{\color{blue}{{\left(c \cdot a\right)}^{2}}}{b} \cdot \left(18 + -12\right) + \left({b}^{3} + \left(-6 \cdot \left(c \cdot \left(a \cdot b\right)\right) + -1 \cdot {b}^{3}\right)\right)}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]

   7. metadata-eval 25.2%

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

   8. +-commutative 25.2%

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

   9. neg-mul-1 25.2%

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

   10. associate-+l+ 96.0%

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

   11. associate-*r* 96.0%

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

   12. neg-mul-1 96.0%

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

   13. distribute-lft1-in 96.0%

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

   14. metadata-eval 96.0%

       \[\leadsto \frac{\frac{\frac{{\left(c \cdot a\right)}^{2}}{b} \cdot 6 + \left(-6 \cdot \left(\left(c \cdot a\right) \cdot b\right) + \color{blue}{0} \cdot {b}^{3}\right)}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]

   15. mul0-lft 96.0%

       \[\leadsto \frac{\frac{\frac{{\left(c \cdot a\right)}^{2}}{b} \cdot 6 + \left(-6 \cdot \left(\left(c \cdot a\right) \cdot b\right) + \color{blue}{0}\right)}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]

9. Simplified 96.0%

   \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(c \cdot a\right)}^{2}}{b} \cdot 6 + \left(-6 \cdot \left(\left(c \cdot a\right) \cdot b\right) + 0\right)}}{c \cdot \left(-a \cdot 6\right) + 3 \cdot \left(b \cdot b\right)}}{2 \cdot a} \]

10. Final simplification 96.0%

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

Alternative 4: 91.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.1%

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

   1. neg-sub0 27.1%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 27.1%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 27.1%

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

   4. neg-mul-1 27.1%

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

   5. associate-*l/ 27.1%

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

   6. *-commutative 27.1%

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

   7. associate-/r* 27.1%

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

   8. /-rgt-identity 27.1%

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

   9. metadata-eval 27.1%

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

3. Simplified 27.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 93.5%

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

   1. distribute-lft-out 93.5%

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

   2. mul-1-neg 93.5%

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

   3. +-commutative 93.5%

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

   4. unpow2 93.5%

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

   5. associate-*l* 93.5%

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

6. Simplified 93.5%

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

7. Final simplification 93.5%

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

Alternative 5: 81.7% 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 27.1%

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

   1. neg-sub0 27.1%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 27.1%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 27.1%

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

   4. neg-mul-1 27.1%

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

   5. associate-*l/ 27.1%

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

   6. *-commutative 27.1%

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

   7. associate-/r* 27.1%

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

   8. /-rgt-identity 27.1%

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

   9. metadata-eval 27.1%

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

3. Simplified 27.1%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf 84.4%

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

   1. associate-*r/ 84.4%

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

   2. neg-mul-1 84.4%

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

6. Simplified 84.4%

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

7. Final simplification 84.4%

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

Alternative 6: 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 27.1%

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

   1. add-cube-cbrt 27.1%

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

   2. pow3 27.1%

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

   3. neg-mul-1 27.1%

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

   4. fma-def 27.1%

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

   5. *-commutative 27.1%

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

   6. *-commutative 27.1%

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

3. Applied egg-rr 27.1%

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

4. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
5. Step-by-step derivation

   1. associate-*r/ 3.2%

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

   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%

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

6. Simplified 3.2%

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

7. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023192 
(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)))