Quadratic roots, medium range

Percentage Accurate: 31.2% → 95.5%

Time: 17.3s

Alternatives: 8

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: 31.2% 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

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

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) (/ a 20.0)) (pow b 7.0)))
    (/ (* 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) / (a / 20.0)) / pow(b, 7.0))) - ((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) / (a / 20.0d0)) / (b ** 7.0d0))) - ((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) / (a / 20.0)) / Math.pow(b, 7.0))) - ((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) / (a / 20.0)) / math.pow(b, 7.0))) - ((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) / Float64(a / 20.0)) / (b ^ 7.0))) - 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) / (a / 20.0)) / (b ^ 7.0))) - ((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[(a / 20.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $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 30.1%

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

   1. *-commutative 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in b around inf 95.9%

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

6. Taylor expanded in c around 0 95.9%

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

8. Simplified 95.9%

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

9. Final simplification 95.9%

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

Alternative 2: 93.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((-2.0 / (pow(b, 5.0) / (pow(a, 2.0) * pow(c, 3.0)))) - (c / b)) - (pow(c, 2.0) * (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 ** 2.0d0) * (c ** 3.0d0)))) - (c / b)) - ((c ** 2.0d0) * (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) / (Math.pow(a, 2.0) * Math.pow(c, 3.0)))) - (c / b)) - (Math.pow(c, 2.0) * (a / Math.pow(b, 3.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.1%

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

   1. *-commutative 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in b around inf 94.4%

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

   1. associate-+r+ 94.4%

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

   2. mul-1-neg 94.4%

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

   3. unsub-neg 94.4%

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

   4. mul-1-neg 94.4%

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

   5. unsub-neg 94.4%

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

   6. associate-*r/ 94.4%

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

   7. associate-/l* 94.4%

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

   8. *-commutative 94.4%

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

   9. associate-/l* 94.4%

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

   10. associate-/r/ 94.4%

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

7. Simplified 94.4%

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

8. Final simplification 94.4%

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

Alternative 3: 93.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -8 - \left(a \cdot c\right) \cdot -4\\ \frac{-0.125 \cdot \frac{{t\_0}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{t\_0}^{3}}{{b}^{5}} + 0.5 \cdot \frac{t\_0}{b}\right)}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* (* a c) -8.0) (* (* a c) -4.0))))
   (/
    (+
     (* -0.125 (/ (pow t_0 2.0) (pow b 3.0)))
     (+ (* 0.0625 (/ (pow t_0 3.0) (pow b 5.0))) (* 0.5 (/ t_0 b))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = ((a * c) * -8.0) - ((a * c) * -4.0);
	return ((-0.125 * (pow(t_0, 2.0) / pow(b, 3.0))) + ((0.0625 * (pow(t_0, 3.0) / pow(b, 5.0))) + (0.5 * (t_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
    real(8) :: t_0
    t_0 = ((a * c) * (-8.0d0)) - ((a * c) * (-4.0d0))
    code = (((-0.125d0) * ((t_0 ** 2.0d0) / (b ** 3.0d0))) + ((0.0625d0 * ((t_0 ** 3.0d0) / (b ** 5.0d0))) + (0.5d0 * (t_0 / b)))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = ((a * c) * -8.0) - ((a * c) * -4.0);
	return ((-0.125 * (Math.pow(t_0, 2.0) / Math.pow(b, 3.0))) + ((0.0625 * (Math.pow(t_0, 3.0) / Math.pow(b, 5.0))) + (0.5 * (t_0 / b)))) / (a * 2.0);
}

Python

def code(a, b, c):
	t_0 = ((a * c) * -8.0) - ((a * c) * -4.0)
	return ((-0.125 * (math.pow(t_0, 2.0) / math.pow(b, 3.0))) + ((0.0625 * (math.pow(t_0, 3.0) / math.pow(b, 5.0))) + (0.5 * (t_0 / b)))) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(a * c) * -8.0) - Float64(Float64(a * c) * -4.0))
	return Float64(Float64(Float64(-0.125 * Float64((t_0 ^ 2.0) / (b ^ 3.0))) + Float64(Float64(0.0625 * Float64((t_0 ^ 3.0) / (b ^ 5.0))) + Float64(0.5 * Float64(t_0 / b)))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = ((a * c) * -8.0) - ((a * c) * -4.0);
	tmp = ((-0.125 * ((t_0 ^ 2.0) / (b ^ 3.0))) + ((0.0625 * ((t_0 ^ 3.0) / (b ^ 5.0))) + (0.5 * (t_0 / b)))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] * -8.0), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(-0.125 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0625 * N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.1%

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

   1. sqr-neg 30.1%

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

   2. +-commutative 30.1%

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

   3. unsub-neg 30.1%

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

   4. sqr-neg 30.1%

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

   5. fma-neg 30.1%

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

   6. distribute-lft-neg-in 30.1%

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

   7. *-commutative 30.1%

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

   8. *-commutative 30.1%

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

   9. distribute-rgt-neg-in 30.1%

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

   10. metadata-eval 30.1%

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

   11. *-commutative 30.1%

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

3. Simplified 30.1%

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

6. Applied egg-rr 30.1%

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

   1. associate--r- 30.0%

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

   2. *-commutative 30.0%

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

   3. count-2 30.0%

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

   4. *-commutative 30.0%

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

8. Simplified 30.0%

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

9. Taylor expanded in b around inf 94.1%

   \[\leadsto \frac{\color{blue}{-0.125 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{-8 \cdot \left(a \cdot c\right) - -4 \cdot \left(a \cdot c\right)}{b}\right)}}{a \cdot 2} \]

10. Final simplification 94.1%

    \[\leadsto \frac{-0.125 \cdot \frac{{\left(\left(a \cdot c\right) \cdot -8 - \left(a \cdot c\right) \cdot -4\right)}^{2}}{{b}^{3}} + \left(0.0625 \cdot \frac{{\left(\left(a \cdot c\right) \cdot -8 - \left(a \cdot c\right) \cdot -4\right)}^{3}}{{b}^{5}} + 0.5 \cdot \frac{\left(a \cdot c\right) \cdot -8 - \left(a \cdot c\right) \cdot -4}{b}\right)}{a \cdot 2} \]
11. Add Preprocessing

Alternative 4: 92.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \left(-1 + \left(1 + {\left(a \cdot c\right)}^{3} \cdot {b}^{-5}\right)\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -4.0 (+ -1.0 (+ 1.0 (* (pow (* a c) 3.0) (pow b -5.0)))))
   (+ (* -2.0 (/ (* a c) b)) (* -2.0 (/ (pow (* a c) 2.0) (pow b 3.0)))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return ((-4.0 * (-1.0 + (1.0 + (pow((a * c), 3.0) * pow(b, -5.0))))) + ((-2.0 * ((a * c) / b)) + (-2.0 * (pow((a * c), 2.0) / pow(b, 3.0))))) / (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 = (((-4.0d0) * ((-1.0d0) + (1.0d0 + (((a * c) ** 3.0d0) * (b ** (-5.0d0)))))) + (((-2.0d0) * ((a * c) / b)) + ((-2.0d0) * (((a * c) ** 2.0d0) / (b ** 3.0d0))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((-4.0 * (-1.0 + (1.0 + (Math.pow((a * c), 3.0) * Math.pow(b, -5.0))))) + ((-2.0 * ((a * c) / b)) + (-2.0 * (Math.pow((a * c), 2.0) / Math.pow(b, 3.0))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return ((-4.0 * (-1.0 + (1.0 + (math.pow((a * c), 3.0) * math.pow(b, -5.0))))) + ((-2.0 * ((a * c) / b)) + (-2.0 * (math.pow((a * c), 2.0) / math.pow(b, 3.0))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-4.0 * Float64(-1.0 + Float64(1.0 + Float64((Float64(a * c) ^ 3.0) * (b ^ -5.0))))) + Float64(Float64(-2.0 * Float64(Float64(a * c) / b)) + Float64(-2.0 * Float64((Float64(a * c) ^ 2.0) / (b ^ 3.0))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-4.0 * (-1.0 + (1.0 + (((a * c) ^ 3.0) * (b ^ -5.0))))) + ((-2.0 * ((a * c) / b)) + (-2.0 * (((a * c) ^ 2.0) / (b ^ 3.0))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-4.0 * N[(-1.0 + N[(1.0 + N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] * N[Power[b, -5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.1%

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

   1. *-commutative 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in b around inf 94.1%

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

   1. expm1-log1p-u 94.1%

      \[\leadsto \frac{-4 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}}\right)\right)} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   2. expm1-udef 92.8%

      \[\leadsto \frac{-4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}}\right)} - 1\right)} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   3. div-inv 92.8%

      \[\leadsto \frac{-4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left({a}^{3} \cdot {c}^{3}\right) \cdot \frac{1}{{b}^{5}}}\right)} - 1\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   4. pow-prod-down 92.8%

      \[\leadsto \frac{-4 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(a \cdot c\right)}^{3}} \cdot \frac{1}{{b}^{5}}\right)} - 1\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   5. pow-flip 92.8%

      \[\leadsto \frac{-4 \cdot \left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{3} \cdot \color{blue}{{b}^{\left(-5\right)}}\right)} - 1\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   6. metadata-eval 92.8%

      \[\leadsto \frac{-4 \cdot \left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{3} \cdot {b}^{\color{blue}{-5}}\right)} - 1\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

7. Applied egg-rr 92.8%

   \[\leadsto \frac{-4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{3} \cdot {b}^{-5}\right)} - 1\right)} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]
8. Step-by-step derivation

   1. sub-neg 92.8%

      \[\leadsto \frac{-4 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{3} \cdot {b}^{-5}\right)} + \left(-1\right)\right)} + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   2. log1p-udef 92.8%

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

   3. rem-exp-log 92.8%

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

   4. metadata-eval 92.8%

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

9. Applied egg-rr 92.8%

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

    1. expm1-log1p-u 92.8%

       \[\leadsto \frac{-4 \cdot \left(\left(1 + {\left(a \cdot c\right)}^{3} \cdot {b}^{-5}\right) + -1\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({a}^{2} \cdot {c}^{2}\right)\right)}}{{b}^{3}}\right)}{a \cdot 2} \]

    2. expm1-udef 89.6%

       \[\leadsto \frac{-4 \cdot \left(\left(1 + {\left(a \cdot c\right)}^{3} \cdot {b}^{-5}\right) + -1\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({a}^{2} \cdot {c}^{2}\right)} - 1}}{{b}^{3}}\right)}{a \cdot 2} \]

    3. pow-prod-down 89.6%

       \[\leadsto \frac{-4 \cdot \left(\left(1 + {\left(a \cdot c\right)}^{3} \cdot {b}^{-5}\right) + -1\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{{\left(a \cdot c\right)}^{2}}\right)} - 1}{{b}^{3}}\right)}{a \cdot 2} \]

11. Applied egg-rr 89.6%

    \[\leadsto \frac{-4 \cdot \left(\left(1 + {\left(a \cdot c\right)}^{3} \cdot {b}^{-5}\right) + -1\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{2}\right)} - 1}}{{b}^{3}}\right)}{a \cdot 2} \]
12. Step-by-step derivation

    1. expm1-def 92.8%

       \[\leadsto \frac{-4 \cdot \left(\left(1 + {\left(a \cdot c\right)}^{3} \cdot {b}^{-5}\right) + -1\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot c\right)}^{2}\right)\right)}}{{b}^{3}}\right)}{a \cdot 2} \]

    2. expm1-log1p 92.8%

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

13. Simplified 92.8%

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

14. Final simplification 92.8%

    \[\leadsto \frac{-4 \cdot \left(-1 + \left(1 + {\left(a \cdot c\right)}^{3} \cdot {b}^{-5}\right)\right) + \left(-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]
15. Add Preprocessing

Alternative 5: 90.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.1%

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

   1. *-commutative 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in b around inf 91.3%

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

   1. mul-1-neg 91.3%

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

   2. unsub-neg 91.3%

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

   3. mul-1-neg 91.3%

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

   4. distribute-neg-frac 91.3%

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

   5. associate-/l* 91.3%

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

   6. associate-/r/ 91.3%

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

7. Simplified 91.3%

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

8. Final simplification 91.3%

   \[\leadsto \frac{-c}{b} - {c}^{2} \cdot \frac{a}{{b}^{3}} \]
9. Add Preprocessing

Alternative 6: 90.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.1%

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

   1. *-commutative 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in b around inf 91.0%

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

   1. add-log-exp 65.4%

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

   2. pow-prod-down 65.4%

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

7. Applied egg-rr 65.4%

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

   1. rem-log-exp 91.0%

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

   2. unpow2 91.0%

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

9. Applied egg-rr 91.0%

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

    1. expm1-log1p-u 90.9%

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

    2. expm1-udef 39.4%

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

    3. associate-/l* 39.4%

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

11. Applied egg-rr 39.4%

    \[\leadsto \frac{-2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a}{\frac{b}{c}}\right)} - 1\right)} + -2 \cdot \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{{b}^{3}}}{a \cdot 2} \]
12. Step-by-step derivation

    1. expm1-def 90.8%

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

    2. expm1-log1p 90.9%

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

    3. associate-/r/ 90.9%

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

13. Simplified 90.9%

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

14. Final simplification 90.9%

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

Alternative 7: 90.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.1%

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

   1. *-commutative 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in b around inf 91.0%

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

   1. add-log-exp 65.4%

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

   2. pow-prod-down 65.4%

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

7. Applied egg-rr 65.4%

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

   1. rem-log-exp 91.0%

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

   2. unpow2 91.0%

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

9. Applied egg-rr 91.0%

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

10. Final simplification 91.0%

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

Alternative 8: 81.4% 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 30.1%

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

   1. *-commutative 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in b around inf 82.3%

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

   1. mul-1-neg 82.3%

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

   2. distribute-neg-frac 82.3%

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

7. Simplified 82.3%

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

8. Final simplification 82.3%

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

Reproduce

herbie shell --seed 2024033 
(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)))