Quadratic roots, medium range

Percentage Accurate: 31.6% → 95.3%

Time: 15.9s

Alternatives: 10

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: 31.6% 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: 95.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((a * ((a * ((-5.0 * ((a * pow(c, 4.0)) / pow(b, 6.0))) + (-2.0 * (pow(c, 3.0) / pow(b, 4.0))))) - ((c / b) * (c / b)))) - 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 ** 6.0d0))) + ((-2.0d0) * ((c ** 3.0d0) / (b ** 4.0d0))))) - ((c / b) * (c / b)))) - 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, 6.0))) + (-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 4.0))))) - ((c / b) * (c / b)))) - c) / b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in b around inf 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in a around 0 96.4%

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

   1. associate-*r/ 96.4%

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

10. Applied egg-rr 96.4%

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

    1. associate-*r/ 96.4%

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

    2. mul-1-neg 96.4%

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

    3. unpow2 96.4%

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

    4. unpow2 96.4%

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

    5. times-frac 96.4%

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

    6. sqr-neg 96.4%

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

    7. distribute-frac-neg2 96.4%

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

    8. distribute-frac-neg2 96.4%

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

    9. unpow2 96.4%

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

12. Simplified 96.4%

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

    1. unpow2 96.4%

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

14. Applied egg-rr 96.4%

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

15. Final simplification 96.4%

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

Alternative 2: 93.6% 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 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in b around inf 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in a around 0 94.6%

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

9. Final simplification 94.6%

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

Alternative 3: 93.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (+
    -1.0
    (* c (- (* -2.0 (/ (* c (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 * ((-2.0 * ((c * 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 * (((-2.0d0) * ((c * (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 * ((-2.0 * ((c * 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 * ((-2.0 * ((c * 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(-2.0 * Float64(Float64(c * (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 * ((-2.0 * ((c * (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[(-2.0 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in b around inf 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in c around 0 94.5%

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

9. Final simplification 94.5%

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

Alternative 4: 93.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (* c (- (* (pow a 2.0) (* -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 * ((pow(a, 2.0) * (-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 * (((a ** 2.0d0) * ((-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 * ((Math.pow(a, 2.0) * (-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 * ((math.pow(a, 2.0) * (-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((a ^ 2.0) * Float64(-2.0 * Float64(c / (b ^ 5.0)))) - Float64(a / (b ^ 3.0)))) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((c * (((a ^ 2.0) * (-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[(N[Power[a, 2.0], $MachinePrecision] * N[(-2.0 * N[(c / N[Power[b, 5.0], $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 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in c around 0 94.3%

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

   1. distribute-lft-in 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-/l* 94.3%

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

   4. associate-*r/ 94.3%

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

   5. neg-mul-1 94.3%

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

7. Applied egg-rr 94.3%

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

   1. distribute-lft-out 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r/ 94.3%

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

   4. associate-*r/ 94.3%

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

   5. *-commutative 94.3%

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

   6. distribute-frac-neg 94.3%

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

   7. unsub-neg 94.3%

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

   8. associate-*l* 94.3%

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

9. Simplified 94.3%

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

10. Final simplification 94.3%

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

Alternative 5: 90.4% 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 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in a around 0 90.7%

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

   1. mul-1-neg 90.7%

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

   2. unsub-neg 90.7%

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

   3. associate-*r/ 90.7%

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

   4. mul-1-neg 90.7%

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

   5. associate-/l* 90.7%

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

7. Simplified 90.7%

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

   1. expm1-log1p-u 57.5%

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

   2. expm1-undefine 34.5%

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

9. Applied egg-rr 34.5%

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

    1. sub-neg 34.5%

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

    2. log1p-undefine 34.5%

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

    3. rem-exp-log 67.7%

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

    4. unsub-neg 67.7%

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

    5. metadata-eval 67.7%

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

11. Simplified 67.7%

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

    1. expm1-log1p-u 54.3%

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

    2. div-inv 54.3%

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

    3. pow-flip 54.3%

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

    4. metadata-eval 54.3%

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

13. Applied egg-rr 54.3%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(1 - c\right) + -1}{b} - a \cdot \left({c}^{2} \cdot {b}^{-3}\right)\right)\right)} \]
14. Step-by-step derivation

    1. expm1-undefine 32.8%

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

    2. sub-neg 32.8%

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

15. Simplified 48.5%

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

16. Taylor expanded in b around inf 90.7%

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

    1. sub-neg 90.7%

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

    2. +-commutative 90.7%

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

    3. mul-1-neg 90.7%

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

    4. distribute-neg-out 90.7%

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

    5. +-commutative 90.7%

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

    6. remove-double-neg 90.7%

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

    7. neg-mul-1 90.7%

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

    8. sub-neg 90.7%

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

18. Simplified 90.7%

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

19. Final simplification 90.7%

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

Alternative 6: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in b around inf 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in c around 0 90.6%

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

9. Final simplification 90.6%

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

Alternative 7: 90.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in c around 0 94.3%

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

   1. log1p-expm1-u 92.8%

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

   2. log1p-undefine 87.8%

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

   3. fma-define 87.8%

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

   4. mul-1-neg 87.8%

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

   5. fmm-undef 87.8%

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

   6. *-commutative 87.8%

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

   7. associate-/l* 87.8%

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

   8. div-inv 87.8%

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

   9. pow-flip 87.8%

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

   10. metadata-eval 87.8%

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

7. Applied egg-rr 87.8%

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

8. Taylor expanded in b around inf 90.5%

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

9. Final simplification 90.5%

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

Alternative 8: 90.2% accurate, 1.0× speedup

Math

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

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(Float64(a * 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[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in c around 0 90.5%

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

   1. associate-*r/ 90.5%

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

   2. neg-mul-1 90.5%

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

   3. distribute-lft-neg-in 90.5%

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

7. Simplified 90.5%

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

8. Final simplification 90.5%

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

Alternative 9: 81.0% 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 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in b around inf 79.8%

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

   1. associate-*r/ 79.8%

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

   2. mul-1-neg 79.8%

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

7. Simplified 79.8%

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

8. Final simplification 79.8%

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

Alternative 10: 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 33.4%

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

   1. *-commutative 33.4%

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

3. Simplified 33.4%

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

5. Taylor expanded in a around 0 90.7%

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

   1. mul-1-neg 90.7%

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

   2. unsub-neg 90.7%

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

   3. associate-*r/ 90.7%

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

   4. mul-1-neg 90.7%

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

   5. associate-/l* 90.7%

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

7. Simplified 90.7%

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

   1. expm1-log1p-u 57.5%

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

   2. expm1-undefine 34.5%

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

9. Applied egg-rr 34.5%

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

    1. sub-neg 34.5%

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

    2. log1p-undefine 34.5%

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

    3. rem-exp-log 67.7%

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

    4. unsub-neg 67.7%

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

    5. metadata-eval 67.7%

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

11. Simplified 67.7%

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

    1. expm1-log1p-u 54.3%

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

    2. div-inv 54.3%

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

    3. pow-flip 54.3%

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

    4. metadata-eval 54.3%

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

13. Applied egg-rr 54.3%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(1 - c\right) + -1}{b} - a \cdot \left({c}^{2} \cdot {b}^{-3}\right)\right)\right)} \]
14. Step-by-step derivation

    1. expm1-undefine 32.8%

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

    2. sub-neg 32.8%

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

15. Simplified 48.5%

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

16. Taylor expanded in c around 0 3.2%

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

17. Final simplification 3.2%

    \[\leadsto 0 \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(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)))