Quadratic roots, narrow range

Percentage Accurate: 55.6% → 99.5%

Time: 20.7s

Alternatives: 12

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.6% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]

9. Taylor expanded in c around 0 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]

12. Taylor expanded in c around inf 0

    \[\leadsto expr\]

14. Simplified 0

    \[\leadsto expr\]
15. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * c) / (b + sqrt(((b * b) + (c * (a * (-4.0d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]

9. Taylor expanded in c around 0 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]
12. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]

9. Taylor expanded in c around 0 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]

13. Applied egg-rr 0

    \[\leadsto expr\]
14. Add Preprocessing

Alternative 4: 90.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ t_1 := b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\\ a \cdot \left(\left(\frac{-2 \cdot t\_0}{t\_1} + \frac{\frac{a \cdot \left(-0.25 \cdot \left(\left(c \cdot t\_0\right) \cdot 20\right)\right)}{b \cdot t\_1}}{b}\right) \cdot a - \frac{\frac{c}{b} \cdot \frac{c}{b}}{b}\right) - \frac{c}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* c c))) (t_1 (* b (* b (* b (* b b))))))
   (-
    (*
     a
     (-
      (*
       (+
        (/ (* -2.0 t_0) t_1)
        (/ (/ (* a (* -0.25 (* (* c t_0) 20.0))) (* b t_1)) b))
       a)
      (/ (* (/ c b) (/ c b)) b)))
    (/ c b))))

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * (b * (b * b)));
	return (a * (((((-2.0 * t_0) / t_1) + (((a * (-0.25 * ((c * t_0) * 20.0))) / (b * t_1)) / b)) * a) - (((c / b) * (c / b)) / b))) - (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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = c * (c * c)
    t_1 = b * (b * (b * (b * b)))
    code = (a * ((((((-2.0d0) * t_0) / t_1) + (((a * ((-0.25d0) * ((c * t_0) * 20.0d0))) / (b * t_1)) / b)) * a) - (((c / b) * (c / b)) / b))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	double t_1 = b * (b * (b * (b * b)));
	return (a * (((((-2.0 * t_0) / t_1) + (((a * (-0.25 * ((c * t_0) * 20.0))) / (b * t_1)) / b)) * a) - (((c / b) * (c / b)) / b))) - (c / b);
}

Python

def code(a, b, c):
	t_0 = c * (c * c)
	t_1 = b * (b * (b * (b * b)))
	return (a * (((((-2.0 * t_0) / t_1) + (((a * (-0.25 * ((c * t_0) * 20.0))) / (b * t_1)) / b)) * a) - (((c / b) * (c / b)) / b))) - (c / b)

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	t_1 = Float64(b * Float64(b * Float64(b * Float64(b * b))))
	return Float64(Float64(a * Float64(Float64(Float64(Float64(Float64(-2.0 * t_0) / t_1) + Float64(Float64(Float64(a * Float64(-0.25 * Float64(Float64(c * t_0) * 20.0))) / Float64(b * t_1)) / b)) * a) - Float64(Float64(Float64(c / b) * Float64(c / b)) / b))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (c * c);
	t_1 = b * (b * (b * (b * b)));
	tmp = (a * (((((-2.0 * t_0) / t_1) + (((a * (-0.25 * ((c * t_0) * 20.0))) / (b * t_1)) / b)) * a) - (((c / b) * (c / b)) / b))) - (c / b);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(N[(N[(N[(N[(-2.0 * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(N[(a * N[(-0.25 * N[(N[(c * t$95$0), $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - N[(N[(N[(c / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in a around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 5: 88.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]

9. Taylor expanded in c around 0 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]

12. Taylor expanded in c around 0 0

    \[\leadsto expr\]

14. Simplified 0

    \[\leadsto expr\]
15. Add Preprocessing

Alternative 6: 88.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]

9. Taylor expanded in c around 0 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]

12. Taylor expanded in c around 0 0

    \[\leadsto expr\]

14. Simplified 0

    \[\leadsto expr\]
15. Add Preprocessing

Alternative 7: 88.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{-0.5 \cdot \frac{b}{c} + a \cdot \left(a \cdot \frac{0.5 \cdot c}{b \cdot \left(b \cdot b\right)} + \frac{0.5}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.5
  (+ (* -0.5 (/ b c)) (* a (+ (* a (/ (* 0.5 c) (* b (* b b)))) (/ 0.5 b))))))

C

double code(double a, double b, double c) {
	return 0.5 / ((-0.5 * (b / c)) + (a * ((a * ((0.5 * c) / (b * (b * b)))) + (0.5 / 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 = 0.5d0 / (((-0.5d0) * (b / c)) + (a * ((a * ((0.5d0 * c) / (b * (b * b)))) + (0.5d0 / b))))
end function

Java

public static double code(double a, double b, double c) {
	return 0.5 / ((-0.5 * (b / c)) + (a * ((a * ((0.5 * c) / (b * (b * b)))) + (0.5 / b))));
}

Python

def code(a, b, c):
	return 0.5 / ((-0.5 * (b / c)) + (a * ((a * ((0.5 * c) / (b * (b * b)))) + (0.5 / b))))

Julia

function code(a, b, c)
	return Float64(0.5 / Float64(Float64(-0.5 * Float64(b / c)) + Float64(a * Float64(Float64(a * Float64(Float64(0.5 * c) / Float64(b * Float64(b * b)))) + Float64(0.5 / b)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.5 / ((-0.5 * (b / c)) + (a * ((a * ((0.5 * c) / (b * (b * b)))) + (0.5 / b))));
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(a * N[(N[(0.5 * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

7. Taylor expanded in a around 0 0

   \[\leadsto expr\]

9. Simplified 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 8: 82.0% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]

9. Taylor expanded in c around 0 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]

12. Taylor expanded in c around 0 0

    \[\leadsto expr\]

14. Simplified 0

    \[\leadsto expr\]
15. Add Preprocessing

Alternative 9: 81.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{-0.5 \cdot \frac{b}{c} + \frac{a \cdot 0.5}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.5 (+ (* -0.5 (/ b c)) (/ (* a 0.5) b))))

C

double code(double a, double b, double c) {
	return 0.5 / ((-0.5 * (b / c)) + ((a * 0.5) / 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 = 0.5d0 / (((-0.5d0) * (b / c)) + ((a * 0.5d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return 0.5 / ((-0.5 * (b / c)) + ((a * 0.5) / b));
}

Python

def code(a, b, c):
	return 0.5 / ((-0.5 * (b / c)) + ((a * 0.5) / b))

Julia

function code(a, b, c)
	return Float64(0.5 / Float64(Float64(-0.5 * Float64(b / c)) + Float64(Float64(a * 0.5) / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.5 / ((-0.5 * (b / c)) + ((a * 0.5) / b));
end

Wolfram

code[a_, b_, c_] := N[(0.5 / N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(N[(a * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

7. Taylor expanded in a around 0 0

   \[\leadsto expr\]

9. Simplified 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 10: 64.2% 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 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

7. Taylor expanded in b around inf 0

   \[\leadsto expr\]

9. Simplified 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 11: 1.6% accurate, 38.7× 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 / 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 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

7. Taylor expanded in b around inf 0

   \[\leadsto expr\]

9. Simplified 0

   \[\leadsto expr\]

11. Applied egg-rr 0

    \[\leadsto expr\]
12. Add Preprocessing

Alternative 12: 1.6% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in b around -inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))