Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.5%

Time: 12.5s

Alternatives: 6

Speedup: 29.0×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * ((pow((a * c), 4.0) / pow(b, 7.0)) * (20.0 / a))) - ((a * 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 = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * ((((a * c) ** 4.0d0) / (b ** 7.0d0)) * (20.0d0 / a))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * ((Math.pow((a * c), 4.0) / Math.pow(b, 7.0)) * (20.0 / a))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) - (c / b));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * ((math.pow((a * c), 4.0) / math.pow(b, 7.0)) * (20.0 / a))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) - (c / b))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * ((((a * c) ^ 4.0) / (b ^ 7.0)) * (20.0 / a))) - ((a * (c ^ 2.0)) / (b ^ 3.0))) - (c / b));
end

Wolfram

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

Derivation

1. Initial program 17.3%

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

   1. *-commutative 17.3%

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

3. Simplified 17.3%

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

4. Taylor expanded in b around inf 97.8%

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

5. Taylor expanded in c around 0 97.8%

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

   1. *-commutative 97.8%

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

7. Simplified 97.8%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{20}{a}\right)}\right)\right) \]

8. Final simplification 97.8%

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

Alternative 2: 96.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

   1. *-commutative 17.3%

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

3. Simplified 17.3%

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

4. Taylor expanded in b around inf 97.2%

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

5. Final simplification 97.2%

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

Alternative 3: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -200000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -200000.0) t_0 (/ (- c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

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) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-200000.0d0)) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -200000.0:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -200000.0)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000.0], t$95$0, N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2e5

   1. Initial program 73.7%

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

   if -2e5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 13.0%

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

      1. *-commutative 13.0%

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

   3. Simplified 13.0%

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

   4. Taylor expanded in b around inf 94.0%

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

      1. mul-1-neg 94.0%

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

   6. Simplified 94.0%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -200000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

   1. *-commutative 17.3%

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

3. Simplified 17.3%

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

4. Taylor expanded in b around inf 95.3%

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

   1. distribute-lft-out 95.3%

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

   2. associate-/l* 95.4%

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

   3. associate-/r/ 95.3%

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

   4. *-commutative 95.3%

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

   5. associate-/l* 95.3%

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

6. Simplified 95.3%

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

7. Taylor expanded in a around 0 95.8%

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

   1. associate-*l/ 95.8%

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

   2. mul-1-neg 95.8%

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

   3. unsub-neg 95.8%

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

   4. associate-*r/ 95.8%

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

   5. neg-mul-1 95.8%

      \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{a}{{b}^{3}} \cdot {c}^{2} \]

   6. associate-*l/ 95.8%

      \[\leadsto \frac{-c}{b} - \color{blue}{\frac{a \cdot {c}^{2}}{{b}^{3}}} \]

   7. associate-*r/ 95.8%

      \[\leadsto \frac{-c}{b} - \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

9. Simplified 95.8%

   \[\leadsto \color{blue}{\frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]

10. Final simplification 95.8%

    \[\leadsto \frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \]

Alternative 5: 94.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

   1. *-commutative 17.3%

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

3. Simplified 17.3%

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

4. Taylor expanded in b around inf 95.3%

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

   1. distribute-lft-out 95.3%

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

   2. associate-/l* 95.4%

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

   3. associate-/r/ 95.3%

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

   4. *-commutative 95.3%

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

   5. associate-/l* 95.3%

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

6. Simplified 95.3%

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

7. Taylor expanded in c around 0 95.3%

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

   1. unpow2 95.3%

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

   2. unpow2 95.3%

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

   3. swap-sqr 95.3%

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

   4. unpow2 95.3%

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

9. Simplified 95.3%

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

    1. *-un-lft-identity 95.3%

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

    2. metadata-eval 95.3%

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

    3. cube-mult 95.3%

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

    4. times-frac 95.3%

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

    5. metadata-eval 95.3%

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

    6. unpow2 95.3%

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

    7. frac-times 95.3%

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

    8. associate-*l/ 95.3%

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

    9. associate-*l/ 95.3%

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

    10. unpow2 95.3%

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

    11. associate-*l/ 95.3%

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

    12. associate-*r/ 95.3%

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

11. Applied egg-rr 95.3%

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

    1. associate-*l/ 95.3%

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

    2. *-lft-identity 95.3%

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

13. Simplified 95.3%

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

14. Final simplification 95.3%

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

Alternative 6: 90.2% 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 17.3%

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

   1. *-commutative 17.3%

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

3. Simplified 17.3%

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

4. Taylor expanded in b around inf 90.9%

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

   1. mul-1-neg 90.9%

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

6. Simplified 90.9%

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

7. Final simplification 90.9%

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

Reproduce

herbie shell --seed 2023336 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))