Quadratic roots, medium range

Percentage Accurate: 31.1% → 95.5%

Time: 13.8s

Alternatives: 7

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: 31.1% 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: 95.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (a * ((a * (((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))) + ((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))))) - ((c ** 2.0d0) / (b ** 3.0d0)))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. *-commutative 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in a around 0 95.7%

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

6. Taylor expanded in c around -inf 95.7%

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

   1. add-log-exp 43.7%

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

   2. mul-1-neg 43.7%

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

8. Applied egg-rr 43.7%

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

9. Taylor expanded in a around 0 95.7%

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

10. Final simplification 95.7%

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

Alternative 2: 95.3% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (-
     (*
      c
      (+
       (* -5.0 (/ (* c (pow a 3.0)) (pow b 7.0)))
       (* -2.0 (/ (pow a 2.0) (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 * ((-5.0 * ((c * pow(a, 3.0)) / pow(b, 7.0))) + (-2.0 * (pow(a, 2.0) / 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 * (((-5.0d0) * ((c * (a ** 3.0d0)) / (b ** 7.0d0))) + ((-2.0d0) * ((a ** 2.0d0) / (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 * ((-5.0 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 7.0))) + (-2.0 * (Math.pow(a, 2.0) / Math.pow(b, 5.0))))) - (a / Math.pow(b, 3.0)))) + (-1.0 / b));
}

Python

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

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * ((c * ((-5.0 * ((c * (a ^ 3.0)) / (b ^ 7.0))) + (-2.0 * ((a ^ 2.0) / (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[(-5.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $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 33.8%

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

   1. *-commutative 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in a around 0 95.7%

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

6. Taylor expanded in c around -inf 95.7%

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

   1. add-log-exp 43.7%

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

   2. mul-1-neg 43.7%

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

8. Applied egg-rr 43.7%

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

9. Taylor expanded in c around 0 95.4%

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

10. Final simplification 95.4%

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

Alternative 3: 94.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. *-commutative 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in a around 0 95.7%

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

6. Taylor expanded in c around 0 93.8%

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

   1. neg-mul-1 93.8%

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

   2. distribute-frac-neg 93.8%

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

   3. associate-*r/ 93.8%

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

   4. distribute-frac-neg 93.8%

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

   5. unsub-neg 93.8%

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

   6. associate-*r/ 93.8%

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

   7. *-commutative 93.8%

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

8. Simplified 93.8%

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

9. Final simplification 93.8%

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

Alternative 4: 93.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. *-commutative 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in c around 0 93.5%

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

   1. expm1-log1p-u 93.5%

      \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({a}^{2} \cdot c\right)\right)}}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right) \]

   2. expm1-undefine 92.6%

      \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({a}^{2} \cdot c\right)} - 1}}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right) \]

   3. *-commutative 92.6%

      \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{c \cdot {a}^{2}}\right)} - 1}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right) \]

7. Applied egg-rr 92.6%

   \[\leadsto c \cdot \left(c \cdot \left(-2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(c \cdot {a}^{2}\right)} - 1}}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right) \]

8. Taylor expanded in a around 0 93.5%

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

   1. pow1 93.5%

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

   2. associate-*r* 93.5%

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

   3. *-commutative 93.5%

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

   4. div-inv 93.5%

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

   5. *-commutative 93.5%

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

   6. pow-flip 93.5%

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

   7. metadata-eval 93.5%

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

   8. pow-flip 93.5%

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

   9. metadata-eval 93.5%

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

10. Applied egg-rr 93.5%

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

    1. unpow1 93.5%

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

    2. *-commutative 93.5%

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

    3. associate-*l* 93.5%

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

12. Simplified 93.5%

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

13. Final simplification 93.5%

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

Alternative 5: 90.9% 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 / 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 33.8%

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

   1. *-commutative 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in c around 0 89.9%

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

   1. associate-*r/ 89.9%

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

   2. neg-mul-1 89.9%

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

   3. distribute-lft-neg-in 89.9%

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

7. Simplified 89.9%

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

8. Taylor expanded in b around inf 90.2%

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

   1. distribute-lft-out 90.2%

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

   2. associate-*r/ 90.2%

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

   3. mul-1-neg 90.2%

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

   4. distribute-neg-frac2 90.2%

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

   5. +-commutative 90.2%

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

   6. associate-/l* 90.2%

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

   7. fma-define 90.2%

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

   8. unpow2 90.2%

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

   9. unpow2 90.2%

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

   10. times-frac 90.2%

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

   11. unpow2 90.2%

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

10. Simplified 90.2%

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

Alternative 6: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. *-commutative 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in c around 0 89.9%

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

   1. associate-*r/ 89.9%

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

   2. neg-mul-1 89.9%

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

   3. distribute-lft-neg-in 89.9%

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

7. Simplified 89.9%

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

8. Taylor expanded in c around 0 89.9%

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

   1. sub-neg 89.9%

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

   2. mul-1-neg 89.9%

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

   3. distribute-neg-out 89.9%

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

   4. +-commutative 89.9%

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

   5. distribute-neg-out 89.9%

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

   6. unsub-neg 89.9%

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

   7. distribute-neg-frac 89.9%

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

   8. metadata-eval 89.9%

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

   9. *-commutative 89.9%

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

   10. associate-*r/ 89.9%

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

10. Simplified 89.9%

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

Alternative 7: 81.5% 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 33.8%

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

   1. *-commutative 33.8%

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

3. Simplified 33.8%

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

5. Taylor expanded in b around inf 79.5%

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

   1. associate-*r/ 79.5%

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

   2. mul-1-neg 79.5%

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

7. Simplified 79.5%

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

8. Final simplification 79.5%

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

Reproduce

herbie shell --seed 2024123 
(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)))