Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.8%

Time: 14.3s

Alternatives: 10

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: 97.8% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (*
    (pow c 2.0)
    (-
     (/ -1.0 (pow b 3.0))
     (* c (* a (- (/ 2.0 (pow b 5.0)) (* -5.0 (/ (* c a) (pow b 7.0)))))))))
  (/ c b)))

C

double code(double a, double b, double c) {
	return (a * (pow(c, 2.0) * ((-1.0 / pow(b, 3.0)) - (c * (a * ((2.0 / pow(b, 5.0)) - (-5.0 * ((c * a) / pow(b, 7.0))))))))) - (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 * ((c ** 2.0d0) * (((-1.0d0) / (b ** 3.0d0)) - (c * (a * ((2.0d0 / (b ** 5.0d0)) - ((-5.0d0) * ((c * a) / (b ** 7.0d0))))))))) - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return (a * (Math.pow(c, 2.0) * ((-1.0 / Math.pow(b, 3.0)) - (c * (a * ((2.0 / Math.pow(b, 5.0)) - (-5.0 * ((c * a) / Math.pow(b, 7.0))))))))) - (c / b);
}

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(a * Float64((c ^ 2.0) * Float64(Float64(-1.0 / (b ^ 3.0)) - Float64(c * Float64(a * Float64(Float64(2.0 / (b ^ 5.0)) - Float64(-5.0 * Float64(Float64(c * a) / (b ^ 7.0))))))))) - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (a * ((c ^ 2.0) * ((-1.0 / (b ^ 3.0)) - (c * (a * ((2.0 / (b ^ 5.0)) - (-5.0 * ((c * a) / (b ^ 7.0))))))))) - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(-1.0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(c * N[(a * N[(N[(2.0 / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(-5.0 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in a around 0 97.2%

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

6. Taylor expanded in c around inf 97.2%

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

7. Taylor expanded in c around 0 97.2%

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

8. Taylor expanded in a around 0 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-*r/ 97.2%

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

   3. metadata-eval 97.2%

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

10. Simplified 97.2%

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

11. Final simplification 97.2%

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

Alternative 2: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in a around 0 96.1%

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

6. Taylor expanded in c around inf 96.1%

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

   1. mul-1-neg 96.1%

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

   2. unsub-neg 96.1%

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

   3. associate-*r/ 96.1%

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

   4. *-commutative 96.1%

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

8. Simplified 96.1%

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

9. Final simplification 96.1%

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

Alternative 3: 97.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in a around 0 97.2%

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

6. Taylor expanded in c around inf 97.2%

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

7. Taylor expanded in c around 0 96.1%

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

8. Final simplification 96.1%

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

Alternative 4: 96.7% 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 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in c around 0 95.8%

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

6. Final simplification 95.8%

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

Alternative 5: 95.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in c around 0 94.0%

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

   1. associate-*r/ 94.0%

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

   2. neg-mul-1 94.0%

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

   3. distribute-lft-neg-in 94.0%

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

7. Simplified 94.0%

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

8. Taylor expanded in b around inf 94.3%

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

   1. distribute-lft-out 94.3%

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

   2. associate-*r/ 94.3%

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

   3. mul-1-neg 94.3%

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

   4. distribute-neg-frac2 94.3%

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

   5. +-commutative 94.3%

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

   6. associate-/l* 94.3%

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

   7. fma-define 94.3%

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

   8. unpow2 94.3%

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

   9. unpow2 94.3%

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

   10. times-frac 94.3%

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

   11. unpow2 94.3%

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

10. Simplified 94.3%

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

Alternative 6: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in c around 0 94.0%

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

   1. associate-*r/ 94.0%

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

   2. neg-mul-1 94.0%

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

   3. distribute-lft-neg-in 94.0%

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

7. Simplified 94.0%

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

8. Taylor expanded in a around inf 93.9%

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

   1. mul-1-neg 93.9%

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

   2. unsub-neg 93.9%

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

   3. mul-1-neg 93.9%

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

   4. distribute-neg-frac2 93.9%

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

   5. distribute-lft-neg-out 93.9%

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

   6. *-commutative 93.9%

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

10. Simplified 93.9%

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

11. Taylor expanded in b around inf 93.8%

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

    1. mul-1-neg 93.8%

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

    2. unpow2 93.8%

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

    3. unpow2 93.8%

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

    4. times-frac 93.8%

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

    5. sqr-neg 93.8%

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

    6. unpow2 93.8%

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

    7. unsub-neg 93.8%

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

    8. mul-1-neg 93.8%

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

    9. distribute-neg-frac2 93.8%

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

    10. distribute-neg-frac2 93.8%

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

13. Simplified 93.8%

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

    1. associate-*r/ 94.1%

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

15. Applied egg-rr 94.1%

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

16. Final simplification 94.1%

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

Alternative 7: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-1 - \frac{c \cdot a}{{b}^{2}}}{b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in c around 0 94.0%

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

   1. associate-*r/ 94.0%

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

   2. neg-mul-1 94.0%

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

   3. distribute-lft-neg-in 94.0%

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

7. Simplified 94.0%

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

8. Taylor expanded in b around inf 94.0%

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

9. Taylor expanded in a around inf 93.9%

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

    1. sub-neg 93.9%

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

    2. mul-1-neg 93.9%

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

    3. distribute-neg-out 93.9%

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

    4. +-commutative 93.9%

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

    5. distribute-rgt-neg-in 93.9%

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

    6. distribute-lft-in 93.9%

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

    7. rgt-mult-inverse 94.0%

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

    8. distribute-neg-in 94.0%

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

    9. metadata-eval 94.0%

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

    10. unsub-neg 94.0%

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

    11. associate-*r/ 94.0%

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

11. Simplified 94.0%

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

12. Final simplification 94.0%

    \[\leadsto c \cdot \frac{-1 - \frac{c \cdot a}{{b}^{2}}}{b} \]
13. Add Preprocessing

Alternative 8: 95.0% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in c around 0 94.0%

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

   1. associate-*r/ 94.0%

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

   2. neg-mul-1 94.0%

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

   3. distribute-lft-neg-in 94.0%

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

7. Simplified 94.0%

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

8. Taylor expanded in a around inf 93.9%

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

   1. mul-1-neg 93.9%

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

   2. unsub-neg 93.9%

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

   3. mul-1-neg 93.9%

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

   4. distribute-neg-frac2 93.9%

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

   5. distribute-lft-neg-out 93.9%

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

   6. *-commutative 93.9%

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

10. Simplified 93.9%

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

11. Taylor expanded in b around inf 93.8%

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

    1. mul-1-neg 93.8%

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

    2. unpow2 93.8%

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

    3. unpow2 93.8%

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

    4. times-frac 93.8%

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

    5. sqr-neg 93.8%

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

    6. unpow2 93.8%

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

    7. unsub-neg 93.8%

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

    8. mul-1-neg 93.8%

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

    9. distribute-neg-frac2 93.8%

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

    10. distribute-neg-frac2 93.8%

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

13. Simplified 93.8%

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

    1. unpow2 93.8%

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

15. Applied egg-rr 93.8%

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

16. Final simplification 93.8%

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

Alternative 9: 90.4% 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(c / Float64(-b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in b around inf 88.4%

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

   1. associate-*r/ 88.4%

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

   2. mul-1-neg 88.4%

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

7. Simplified 88.4%

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

8. Final simplification 88.4%

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

Alternative 10: 3.3% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 20.5%

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

   1. *-commutative 20.5%

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

3. Simplified 20.5%

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

5. Taylor expanded in c around 0 94.0%

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

   1. associate-*r/ 94.0%

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

   2. neg-mul-1 94.0%

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

   3. distribute-lft-neg-in 94.0%

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

7. Simplified 94.0%

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

8. Taylor expanded in a around 0 88.1%

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

   1. expm1-log1p-u 74.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)\right)} \]

   2. expm1-undefine 18.3%

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

10. Applied egg-rr 18.3%

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

    1. sub-neg 18.3%

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

    2. log1p-undefine 18.3%

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

    3. rem-exp-log 31.7%

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

    4. associate-*r/ 31.8%

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

    5. *-commutative 31.8%

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

    6. mul-1-neg 31.8%

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

    7. distribute-neg-frac 31.8%

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

    8. unsub-neg 31.8%

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

    9. metadata-eval 31.8%

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

12. Simplified 31.8%

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

13. Taylor expanded in c around 0 3.3%

    \[\leadsto \color{blue}{1} + -1 \]

14. Final simplification 3.3%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024119 
(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)))