Quadratic roots, wide range

Percentage Accurate: 17.7% → 97.7%

Time: 11.3s

Alternatives: 9

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: 17.7% 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 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.8%

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

5. Taylor expanded in a around 0 97.8%

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

6. Taylor expanded in c around -inf 97.8%

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

   1. mul-1-neg 97.8%

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

8. Applied egg-rr 97.8%

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

9. Final simplification 97.8%

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

Alternative 2: 97.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (*
   a
   (-
    (* (* a -2.0) (* (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 * (((a * -2.0) * (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 * (((a * (-2.0d0)) * ((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 * (((a * -2.0) * (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 * (((a * -2.0) * (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(a * -2.0) * Float64((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 * (((a * -2.0) * ((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[(a * -2.0), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[b, -5.0], $MachinePrecision]), $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 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.8%

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

5. Taylor expanded in a around 0 97.8%

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

6. Taylor expanded in a around 0 97.2%

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

   1. mul-1-neg 97.2%

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

   2. unsub-neg 97.2%

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

   3. associate-*r/ 97.2%

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

8. Simplified 97.2%

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

   1. *-un-lft-identity 97.2%

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

   2. +-commutative 97.2%

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

   3. fma-define 97.2%

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

   4. div-inv 97.2%

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

   5. associate-*r* 97.2%

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

   6. pow-flip 97.2%

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

   7. metadata-eval 97.2%

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

   8. div-inv 97.2%

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

   9. pow-flip 97.2%

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

   10. metadata-eval 97.2%

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

   11. mul-1-neg 97.2%

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

10. Applied egg-rr 97.2%

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

    1. *-lft-identity 97.2%

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

    2. fmm-undef 97.2%

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

    3. associate-*l* 97.2%

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

    4. *-commutative 97.2%

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

12. Simplified 97.2%

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

Alternative 3: 96.6% 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 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 96.7%

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

6. Final simplification 96.7%

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

Alternative 4: 95.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.8%

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

5. Taylor expanded in a around 0 95.9%

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

   1. mul-1-neg 95.9%

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

   2. unsub-neg 95.9%

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

   3. mul-1-neg 95.9%

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

   4. distribute-neg-frac2 95.9%

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

   5. associate-/l* 95.9%

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

7. Simplified 95.9%

   \[\leadsto \color{blue}{\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}} \]
8. 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 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 95.5%

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

   1. associate-*r/ 95.5%

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

   2. neg-mul-1 95.5%

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

   3. distribute-rgt-neg-in 95.5%

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

7. Simplified 95.5%

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

8. Taylor expanded in b around 0 95.3%

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

   1. distribute-lft-out 95.3%

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

10. Simplified 95.3%

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

11. Taylor expanded in b around inf 95.9%

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

    1. distribute-lft-out 95.9%

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

    2. associate-*r/ 95.9%

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

    3. mul-1-neg 95.9%

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

    4. distribute-neg-frac2 95.9%

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

    5. +-commutative 95.9%

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

    6. associate-/l* 95.9%

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

    7. fma-define 95.9%

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

    8. unpow2 95.9%

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

    9. unpow2 95.9%

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

    10. times-frac 95.9%

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

    11. unpow2 95.9%

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

13. Simplified 95.9%

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

Alternative 6: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 95.5%

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

   1. associate-*r/ 95.5%

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

   2. neg-mul-1 95.5%

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

   3. distribute-rgt-neg-in 95.5%

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

7. Simplified 95.5%

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

8. Taylor expanded in b around 0 95.3%

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

   1. distribute-lft-out 95.3%

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

10. Simplified 95.3%

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

11. Taylor expanded in c around 0 95.5%

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

    1. associate-*r/ 95.5%

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

    2. unpow3 95.5%

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

    3. unpow2 95.5%

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

    4. associate-/r* 95.5%

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

    5. associate-*r/ 95.5%

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

    6. div-sub 95.4%

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

    7. fmm-def 95.4%

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

    8. associate-*r/ 95.4%

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

    9. metadata-eval 95.4%

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

    10. *-lft-identity 95.4%

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

    11. associate-*l/ 95.4%

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

    12. associate-*r* 95.5%

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

    13. associate-*r/ 95.8%

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

    14. *-rgt-identity 95.8%

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

    15. fma-undefine 95.8%

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

    16. neg-mul-1 95.8%

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

    17. +-commutative 95.8%

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

13. Simplified 95.8%

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

Alternative 7: 95.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.8%

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

5. Taylor expanded in b around inf 95.4%

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

   1. distribute-lft-out 95.4%

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

   2. *-commutative 95.4%

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

7. Simplified 95.4%

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

8. Taylor expanded in c around 0 95.5%

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

   1. sub-neg 95.5%

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

   2. associate-*r/ 95.5%

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

   3. neg-mul-1 95.5%

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

   4. distribute-rgt-neg-in 95.5%

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

   5. associate-*r/ 95.5%

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

   6. +-commutative 95.5%

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

   7. distribute-frac-neg 95.5%

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

   8. distribute-rgt-neg-in 95.5%

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

   9. associate-/l* 95.5%

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

   10. unsub-neg 95.5%

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

   11. distribute-neg-frac 95.5%

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

   12. metadata-eval 95.5%

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

   13. associate-/l* 95.5%

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

10. Simplified 95.5%

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

Alternative 8: 95.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.8%

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

5. Taylor expanded in c around 0 95.5%

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

   1. associate-*r/ 95.5%

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

   2. neg-mul-1 95.5%

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

   3. distribute-rgt-neg-in 95.5%

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

7. Simplified 95.5%

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

8. Taylor expanded in b around inf 95.4%

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

   1. unpow2 95.4%

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

10. Applied egg-rr 95.4%

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

11. Final simplification 95.4%

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

Alternative 9: 90.5% 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 16.7%

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

   1. *-commutative 16.7%

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

3. Simplified 16.8%

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

5. Taylor expanded in b around inf 91.1%

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

   1. associate-*r/ 91.1%

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

   2. mul-1-neg 91.1%

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

7. Simplified 91.1%

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

8. Final simplification 91.1%

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

Reproduce

herbie shell --seed 2024180 
(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)))