Quadratic roots, wide range

Percentage Accurate: 18.2% → 97.7%

Time: 16.7s

Alternatives: 8

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: 18.2% 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: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.7%

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

   1. *-commutative 14.7%

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

3. Simplified 14.7%

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

5. Taylor expanded in a around 0 97.5%

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

6. Taylor expanded in c around -inf 97.5%

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

7. Final simplification 97.5%

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

Alternative 2: 96.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.7%

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

   1. *-commutative 14.7%

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

3. Simplified 14.7%

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

5. Taylor expanded in a around 0 97.5%

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

6. Taylor expanded in a around 0 96.8%

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

   1. mul-1-neg 96.8%

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

   2. unsub-neg 96.8%

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

   3. associate-*r/ 96.8%

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

8. Simplified 96.8%

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

9. Final simplification 96.8%

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

Alternative 3: 96.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * ((c * (((-2.0d0) * ((c * (a ** 2.0d0)) / (b ** 5.0d0))) - (a / (b ** 3.0d0)))) + ((-1.0d0) / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.7%

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

   1. *-commutative 14.7%

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

3. Simplified 14.7%

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

5. Taylor expanded in c around 0 96.4%

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

6. Final simplification 96.4%

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

Alternative 4: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.7%

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

   1. *-commutative 14.7%

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

3. Simplified 14.7%

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

5. Taylor expanded in b around inf 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

   3. mul-1-neg 95.7%

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

7. Simplified 95.7%

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

   1. associate-/l* 95.7%

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

9. Applied egg-rr 95.7%

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

    1. unpow2 95.7%

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

    2. unpow2 95.7%

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

    3. times-frac 95.7%

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

    4. sqr-neg 95.7%

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

    5. distribute-frac-neg 95.7%

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

    6. neg-mul-1 95.7%

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

    7. *-commutative 95.7%

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

    8. associate-*r/ 95.7%

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

    9. distribute-frac-neg 95.7%

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

    10. neg-mul-1 95.7%

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

    11. *-commutative 95.7%

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

    12. associate-*r/ 95.7%

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

    13. unpow1 95.7%

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

    14. pow-plus 95.7%

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

    15. associate-*r/ 95.7%

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

    16. *-commutative 95.7%

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

    17. neg-mul-1 95.7%

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

    18. distribute-frac-neg 95.7%

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

    19. distribute-neg-frac2 95.7%

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

    20. metadata-eval 95.7%

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

11. Simplified 95.7%

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

12. Final simplification 95.7%

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

Alternative 5: 95.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((((c * a) / b) - b) / c);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / ((((c * a) / b) - b) / c)
end function

Java

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

Python

def code(a, b, c):
	return 1.0 / ((((c * a) / b) - b) / c)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.7%

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

   1. *-commutative 14.7%

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

3. Simplified 14.7%

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

5. Taylor expanded in b around inf 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

   3. mul-1-neg 95.7%

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

7. Simplified 95.7%

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

   1. clear-num 95.3%

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

   2. inv-pow 95.3%

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

   3. associate-/l* 95.3%

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

9. Applied egg-rr 95.3%

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

    1. unpow-1 95.3%

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

    2. unpow2 95.3%

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

    3. unpow2 95.3%

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

    4. times-frac 95.3%

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

    5. sqr-neg 95.3%

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

    6. distribute-frac-neg 95.3%

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

    7. neg-mul-1 95.3%

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

    8. *-commutative 95.3%

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

    9. associate-*r/ 95.3%

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

    10. distribute-frac-neg 95.3%

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

    11. neg-mul-1 95.3%

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

    12. *-commutative 95.3%

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

    13. associate-*r/ 95.3%

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

    14. unpow1 95.3%

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

    15. pow-plus 95.3%

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

    16. associate-*r/ 95.3%

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

    17. *-commutative 95.3%

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

    18. neg-mul-1 95.3%

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

    19. distribute-frac-neg 95.3%

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

    20. distribute-neg-frac2 95.3%

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

    21. metadata-eval 95.3%

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

11. Simplified 95.3%

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

12. Taylor expanded in c around 0 95.5%

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

13. Final simplification 95.5%

    \[\leadsto \frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}} \]
14. Add Preprocessing

Alternative 6: 95.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / ((a / b) - (b / c))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.7%

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

   1. *-commutative 14.7%

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

3. Simplified 14.7%

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

5. Taylor expanded in b around inf 95.7%

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

   1. mul-1-neg 95.7%

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

   2. unsub-neg 95.7%

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

   3. mul-1-neg 95.7%

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

7. Simplified 95.7%

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

   1. clear-num 95.3%

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

   2. inv-pow 95.3%

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

   3. associate-/l* 95.3%

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

9. Applied egg-rr 95.3%

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

    1. unpow-1 95.3%

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

    2. unpow2 95.3%

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

    3. unpow2 95.3%

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

    4. times-frac 95.3%

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

    5. sqr-neg 95.3%

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

    6. distribute-frac-neg 95.3%

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

    7. neg-mul-1 95.3%

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

    8. *-commutative 95.3%

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

    9. associate-*r/ 95.3%

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

    10. distribute-frac-neg 95.3%

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

    11. neg-mul-1 95.3%

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

    12. *-commutative 95.3%

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

    13. associate-*r/ 95.3%

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

    14. unpow1 95.3%

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

    15. pow-plus 95.3%

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

    16. associate-*r/ 95.3%

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

    17. *-commutative 95.3%

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

    18. neg-mul-1 95.3%

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

    19. distribute-frac-neg 95.3%

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

    20. distribute-neg-frac2 95.3%

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

    21. metadata-eval 95.3%

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

11. Simplified 95.3%

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

12. Taylor expanded in a around 0 95.5%

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

    1. +-commutative 95.5%

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

    2. mul-1-neg 95.5%

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

    3. unsub-neg 95.5%

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

14. Simplified 95.5%

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

15. Final simplification 95.5%

    \[\leadsto \frac{1}{\frac{a}{b} - \frac{b}{c}} \]
16. Add Preprocessing

Alternative 7: 90.1% 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 14.7%

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

   1. *-commutative 14.7%

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

3. Simplified 14.7%

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

5. Taylor expanded in b around inf 92.4%

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

   1. associate-*r/ 92.4%

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

   2. mul-1-neg 92.4%

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

7. Simplified 92.4%

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

8. Final simplification 92.4%

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

Alternative 8: 3.3% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 14.7%

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

   1. *-commutative 14.7%

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

3. Simplified 14.7%

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

   1. flip3-- 14.6%

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

   2. sqrt-div 14.8%

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

   3. pow2 14.8%

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

   4. pow-pow 15.2%

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

   5. metadata-eval 15.2%

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

   6. associate-*l* 15.2%

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

   7. unpow-prod-down 15.2%

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

   8. metadata-eval 15.2%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

6. Applied egg-rr 15.1%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]
7. Step-by-step derivation

   1. *-un-lft-identity 15.1%

      \[\leadsto \color{blue}{1 \cdot \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}{a \cdot 2}} \]

8. Applied egg-rr 14.7%

   \[\leadsto \color{blue}{1 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(c \cdot a\right), \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right), {b}^{4}\right)}}\right)}{2 \cdot a}} \]
9. Step-by-step derivation

   1. associate-*r/ 14.7%

      \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(c \cdot a\right), \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right), {b}^{4}\right)}}\right)}{2 \cdot a}} \]

   2. *-commutative 14.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(c \cdot a\right), \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right), {b}^{4}\right)}}\right) \cdot 1}}{2 \cdot a} \]

   3. associate-*r/ 14.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(c \cdot a\right), \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right), {b}^{4}\right)}}\right) \cdot \frac{1}{2 \cdot a}} \]

   4. *-commutative 14.7%

      \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(c \cdot a\right), \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right), {b}^{4}\right)}}\right)} \]

   5. associate-/r* 14.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(c \cdot a\right), \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right), {b}^{4}\right)}}\right) \]

   6. metadata-eval 14.7%

      \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -64 \cdot {\left(c \cdot a\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(c \cdot a\right), \mathsf{fma}\left(b, b, 4 \cdot \left(c \cdot a\right)\right), {b}^{4}\right)}}\right) \]

10. Simplified 14.7%

    \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\frac{{b}^{6} + -64 \cdot {\left(a \cdot c\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(a \cdot c\right), \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right), {b}^{4}\right)}}\right)} \]

11. Taylor expanded in a around 0 3.3%

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

    1. associate-*r/ 3.3%

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

    2. distribute-rgt1-in 3.3%

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

    3. metadata-eval 3.3%

       \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 3.3%

       \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.3%

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

13. Simplified 3.3%

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

14. Final simplification 3.3%

    \[\leadsto \frac{0}{a} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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)))