Quadratic roots, medium range

Percentage Accurate: 31.2% → 95.4%

Time: 16.2s

Alternatives: 11

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \frac{-{c}^{2}}{{b}^{2}}\right) - c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -2.0
   (* (pow a 2.0) (/ (pow c 3.0) (pow b 4.0)))
   (-
    (fma
     -0.25
     (* (/ (* (pow a 4.0) (pow c 4.0)) a) (/ 20.0 (pow b 6.0)))
     (* a (/ (- (pow c 2.0)) (pow b 2.0))))
    c))
  b))

C

double code(double a, double b, double c) {
	return fma(-2.0, (pow(a, 2.0) * (pow(c, 3.0) / pow(b, 4.0))), (fma(-0.25, (((pow(a, 4.0) * pow(c, 4.0)) / a) * (20.0 / pow(b, 6.0))), (a * (-pow(c, 2.0) / pow(b, 2.0)))) - c)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(-2.0, Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 4.0))), Float64(fma(-0.25, Float64(Float64(Float64((a ^ 4.0) * (c ^ 4.0)) / a) * Float64(20.0 / (b ^ 6.0))), Float64(a * Float64(Float64(-(c ^ 2.0)) / (b ^ 2.0)))) - c)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[((-N[Power[c, 2.0], $MachinePrecision]) / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in b around inf 95.6%

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

7. Simplified 95.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \left(-\frac{{c}^{2}}{{b}^{2}}\right)\right) - c\right)}{b}} \]

8. Final simplification 95.6%

   \[\leadsto \frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \frac{-{c}^{2}}{{b}^{2}}\right) - c\right)}{b} \]
9. Add Preprocessing

Alternative 2: 95.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in b around inf 95.6%

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

7. Simplified 95.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \left(-\frac{{c}^{2}}{{b}^{2}}\right)\right) - c\right)}{b}} \]

8. Taylor expanded in a around 0 95.6%

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

   1. unpow2 95.6%

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

10. Applied egg-rr 95.6%

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

    1. mul-1-neg 95.6%

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

    2. distribute-frac-neg 95.6%

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

12. Applied egg-rr 95.6%

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

13. Final simplification 95.6%

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

Alternative 3: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(-1 + c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+
    -1.0
    (*
     c
     (-
      (*
       c
       (+
        (* -5.0 (/ (* c (pow a 3.0)) (pow b 6.0)))
        (* -2.0 (/ (pow a 2.0) (pow b 4.0)))))
      (/ a (pow b 2.0))))))
  b))

C

double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((c * ((-5.0 * ((c * pow(a, 3.0)) / pow(b, 6.0))) + (-2.0 * (pow(a, 2.0) / pow(b, 4.0))))) - (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 * ((c * (((-5.0d0) * ((c * (a ** 3.0d0)) / (b ** 6.0d0))) + ((-2.0d0) * ((a ** 2.0d0) / (b ** 4.0d0))))) - (a / (b ** 2.0d0)))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * (-1.0 + (c * ((c * ((-5.0 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 6.0))) + (-2.0 * (Math.pow(a, 2.0) / Math.pow(b, 4.0))))) - (a / Math.pow(b, 2.0)))))) / b;
}

Python

def code(a, b, c):
	return (c * (-1.0 + (c * ((c * ((-5.0 * ((c * math.pow(a, 3.0)) / math.pow(b, 6.0))) + (-2.0 * (math.pow(a, 2.0) / math.pow(b, 4.0))))) - (a / math.pow(b, 2.0)))))) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(-1.0 + Float64(c * Float64(Float64(c * Float64(Float64(-5.0 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 6.0))) + Float64(-2.0 * Float64((a ^ 2.0) / (b ^ 4.0))))) - Float64(a / (b ^ 2.0)))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (-1.0 + (c * ((c * ((-5.0 * ((c * (a ^ 3.0)) / (b ^ 6.0))) + (-2.0 * ((a ^ 2.0) / (b ^ 4.0))))) - (a / (b ^ 2.0)))))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(-1.0 + N[(c * N[(N[(c * N[(N[(-5.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in b around inf 95.6%

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

7. Simplified 95.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \left(-\frac{{c}^{2}}{{b}^{2}}\right)\right) - c\right)}{b}} \]

8. Taylor expanded in c around 0 95.5%

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

9. Final simplification 95.5%

   \[\leadsto \frac{c \cdot \left(-1 + c \cdot \left(c \cdot \left(-5 \cdot \frac{c \cdot {a}^{3}}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right)\right)}{b} \]
10. Add Preprocessing

Alternative 4: 95.2% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (*
    c
    (-
     (*
      c
      (+
       (* -5.0 (/ (* c (pow a 3.0)) (pow b 7.0)))
       (* -2.0 (/ (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 * ((c * ((-5.0 * ((c * pow(a, 3.0)) / pow(b, 7.0))) + (-2.0 * (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 * ((c * (((-5.0d0) * ((c * (a ** 3.0d0)) / (b ** 7.0d0))) + ((-2.0d0) * ((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 * ((c * ((-5.0 * ((c * Math.pow(a, 3.0)) / Math.pow(b, 7.0))) + (-2.0 * (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 * ((c * ((-5.0 * ((c * math.pow(a, 3.0)) / math.pow(b, 7.0))) + (-2.0 * (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(c * Float64(Float64(-5.0 * Float64(Float64(c * (a ^ 3.0)) / (b ^ 7.0))) + Float64(-2.0 * Float64((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 * ((c * ((-5.0 * ((c * (a ^ 3.0)) / (b ^ 7.0))) + (-2.0 * ((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[(c * N[(N[(-5.0 * N[(N[(c * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $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 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in b around inf 95.6%

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

7. Simplified 95.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \left(-\frac{{c}^{2}}{{b}^{2}}\right)\right) - c\right)}{b}} \]

8. Taylor expanded in c around 0 95.3%

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

9. Final simplification 95.3%

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

Alternative 5: 93.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 4.0))) - (pow(c, 2.0) / pow(b, 2.0)))) - 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 ** 4.0d0))) - ((c ** 2.0d0) / (b ** 2.0d0)))) - 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, 4.0))) - (Math.pow(c, 2.0) / Math.pow(b, 2.0)))) - c) / b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in b around inf 95.6%

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

7. Simplified 95.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, {a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.25, \frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{20}{{b}^{6}}, a \cdot \left(-\frac{{c}^{2}}{{b}^{2}}\right)\right) - c\right)}{b}} \]

8. Taylor expanded in a around 0 93.8%

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

9. Final simplification 93.8%

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

Alternative 6: 93.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. *-commutative 33.0%

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

   2. +-commutative 33.0%

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

   3. sqr-neg 33.0%

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

   4. unsub-neg 33.0%

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

   5. sqr-neg 33.0%

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

   6. fma-neg 33.1%

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

   7. distribute-lft-neg-in 33.1%

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

   8. *-commutative 33.1%

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

   9. *-commutative 33.1%

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

   10. distribute-rgt-neg-in 33.1%

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

   11. metadata-eval 33.1%

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

3. Simplified 33.1%

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

5. Taylor expanded in a around inf 32.8%

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

6. Taylor expanded in b around inf 93.8%

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

   1. associate-+r+ 93.8%

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

   2. mul-1-neg 93.8%

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

   3. unsub-neg 93.8%

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

   4. mul-1-neg 93.8%

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

   5. unsub-neg 93.8%

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

   6. associate-/l* 93.8%

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

   7. associate-/l* 93.8%

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

   8. unpow2 93.8%

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

   9. unpow2 93.8%

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

   10. times-frac 93.8%

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

   11. unpow2 93.8%

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

8. Simplified 93.8%

   \[\leadsto \color{blue}{\frac{\left(-2 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}\right) - c\right) - a \cdot {\left(\frac{c}{b}\right)}^{2}}{b}} \]
9. Add Preprocessing

Alternative 7: 93.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in a around 0 93.8%

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

6. Taylor expanded in c around inf 93.8%

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

   1. associate-*r/ 93.8%

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

   2. associate-/r* 93.8%

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

8. Simplified 93.8%

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

9. Final simplification 93.8%

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

Alternative 8: 93.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (- (* -2.0 (/ (* (pow a 2.0) c) (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 * ((pow(a, 2.0) * c) / 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) * (((a ** 2.0d0) * c) / (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 * ((Math.pow(a, 2.0) * c) / 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 * ((math.pow(a, 2.0) * c) / 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((a ^ 2.0) * c) / (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 * (((a ^ 2.0) * c) / (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[(N[Power[a, 2.0], $MachinePrecision] * c), $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 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in c around 0 93.6%

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

6. Final simplification 93.6%

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

Alternative 9: 90.8% 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(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 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in c around 0 89.9%

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

   1. associate-*r/ 89.9%

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

   2. neg-mul-1 89.9%

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

   3. distribute-lft-neg-in 89.9%

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

7. Simplified 89.9%

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

8. Taylor expanded in b around inf 90.2%

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

   1. neg-mul-1 90.2%

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

   2. +-commutative 90.2%

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

   3. sub-neg 90.2%

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

   4. associate-*r/ 90.2%

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

   5. mul-1-neg 90.2%

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

10. Simplified 90.2%

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

11. Taylor expanded in a around 0 90.2%

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

    1. neg-mul-1 90.2%

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

    2. associate-/l* 90.2%

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

    3. distribute-lft-neg-in 90.2%

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

    4. unpow2 90.2%

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

    5. unpow2 90.2%

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

    6. times-frac 90.2%

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

    7. unpow1 90.2%

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

    8. pow-plus 90.2%

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

    9. metadata-eval 90.2%

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

13. Simplified 90.2%

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

14. Final simplification 90.2%

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

Alternative 10: 81.4% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(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 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in b around inf 79.7%

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

   1. associate-*r/ 79.7%

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

   2. mul-1-neg 79.7%

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

7. Simplified 79.7%

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

8. Final simplification 79.7%

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

Alternative 11: 3.2% 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 33.0%

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

   1. *-commutative 33.0%

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

3. Simplified 33.0%

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

5. Taylor expanded in b around inf 79.7%

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

   1. associate-*r/ 79.7%

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

   2. mul-1-neg 79.7%

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

7. Simplified 79.7%

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

   1. distribute-frac-neg 79.7%

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

   2. mul-1-neg 79.7%

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

   3. expm1-log1p-u 72.0%

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

   4. expm1-undefine 32.2%

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

   5. mul-1-neg 32.2%

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

   6. distribute-frac-neg 32.2%

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

9. Applied egg-rr 32.2%

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

    1. sub-neg 32.2%

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

    2. log1p-undefine 32.2%

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

    3. rem-exp-log 39.8%

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

    4. distribute-frac-neg 39.8%

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

    5. unsub-neg 39.8%

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

    6. metadata-eval 39.8%

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

11. Simplified 39.8%

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

12. Taylor expanded in c around 0 3.2%

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

13. Final simplification 3.2%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024118 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))