Quadratic roots, narrow range

Percentage Accurate: 55.4% → 90.7%

Time: 18.1s

Alternatives: 14

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.4% 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: 90.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

   1. *-commutative 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in c around 0 91.2%

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

   1. fma-neg 91.2%

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

7. Simplified 91.2%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}}}{b} \cdot \frac{20}{a}\right)\right)\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right)} \]

8. Taylor expanded in b around -inf 91.3%

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

9. Final simplification 91.3%

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

Alternative 2: 90.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

   1. *-commutative 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in c around 0 91.2%

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

   1. fma-neg 91.2%

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

7. Simplified 91.2%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}}}{b} \cdot \frac{20}{a}\right)\right)\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right)} \]

8. Taylor expanded in b around -inf 91.3%

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

   1. associate-/l* 91.3%

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

10. Applied egg-rr 91.3%

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

    1. unpow2 91.3%

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

    2. unpow2 91.3%

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

    3. times-frac 91.3%

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

    4. unpow2 91.3%

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

12. Simplified 91.3%

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

13. Final simplification 91.3%

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

Alternative 3: 90.7% accurate, 0.2× speedup

Math

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

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)))))
    (/ (pow c 2.0) (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))))) - (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 * (((-5.0d0) * ((a * (c ** 4.0d0)) / (b ** 7.0d0))) + ((-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 * ((-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))))) - (Math.pow(c, 2.0) / 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))))) - (math.pow(c, 2.0) / 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((c ^ 2.0) / (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 ^ 2.0) / (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[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

   1. *-commutative 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in c around 0 91.2%

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

   1. fma-neg 91.2%

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

7. Simplified 91.2%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}}}{b} \cdot \frac{20}{a}\right)\right)\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right)} \]

8. Taylor expanded in a around 0 91.3%

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

9. Final simplification 91.3%

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

Alternative 4: 90.5% 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 57.2%

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

   1. *-commutative 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in c around 0 91.2%

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

   1. fma-neg 91.2%

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

7. Simplified 91.2%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}}}{b} \cdot \frac{20}{a}\right)\right)\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right)} \]

8. Taylor expanded in c around 0 91.2%

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

9. Final simplification 91.2%

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

Alternative 5: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right) + 2 \cdot \left({c}^{3} \cdot \frac{{a}^{2}}{{b}^{4}}\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (/
    (+
     (fma a (pow (/ c b) 2.0) c)
     (* 2.0 (* (pow c 3.0) (/ (pow a 2.0) (pow b 4.0)))))
    (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (fma(a, pow((c / b), 2.0), c) + (2.0 * (pow(c, 3.0) * (pow(a, 2.0) / pow(b, 4.0))))) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(a, (Float64(c / b) ^ 2.0), c) + Float64(2.0 * Float64((c ^ 3.0) * Float64((a ^ 2.0) / (b ^ 4.0))))) / Float64(-b));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 1

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. +-commutative 82.9%

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

      3. sqr-neg 82.9%

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

      4. unsub-neg 82.9%

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

      5. sqr-neg 82.9%

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

      6. fma-neg 83.1%

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

      7. distribute-lft-neg-in 83.1%

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

      8. *-commutative 83.1%

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

      9. *-commutative 83.1%

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

      10. distribute-rgt-neg-in 83.1%

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

      11. metadata-eval 83.1%

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

   3. Simplified 83.1%

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

   if 1 < b

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in c around 0 93.4%

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

      1. fma-neg 93.4%

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

   7. Simplified 93.4%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(c, c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2}}{{b}^{5}}, -0.25 \cdot \left(c \cdot \left(\frac{\frac{{a}^{4}}{{b}^{6}}}{b} \cdot \frac{20}{a}\right)\right)\right) - \frac{a}{{b}^{3}}, \frac{-1}{b}\right)} \]

   8. Taylor expanded in b around -inf 90.8%

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

      1. mul-1-neg 90.8%

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

      2. distribute-neg-frac2 90.8%

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

   10. Simplified 90.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right) + 2 \cdot \left({c}^{3} \cdot \frac{{a}^{2}}{{b}^{4}}\right)}{-b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right) + 2 \cdot \left({c}^{3} \cdot \frac{{a}^{2}}{{b}^{4}}\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{3} \cdot \left(\left(a \cdot -2\right) \cdot {b}^{-5} - \frac{{b}^{-3}}{c}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (-
    (* a (* (pow c 3.0) (- (* (* a -2.0) (pow b -5.0)) (/ (pow b -3.0) c))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (a * (pow(c, 3.0) * (((a * -2.0) * pow(b, -5.0)) - (pow(b, -3.0) / c)))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(a * Float64((c ^ 3.0) * Float64(Float64(Float64(a * -2.0) * (b ^ -5.0)) - Float64((b ^ -3.0) / c)))) - Float64(c / b));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 1

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. +-commutative 82.9%

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

      3. sqr-neg 82.9%

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

      4. unsub-neg 82.9%

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

      5. sqr-neg 82.9%

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

      6. fma-neg 83.1%

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

      7. distribute-lft-neg-in 83.1%

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

      8. *-commutative 83.1%

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

      9. *-commutative 83.1%

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

      10. distribute-rgt-neg-in 83.1%

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

      11. metadata-eval 83.1%

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

   3. Simplified 83.1%

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

   if 1 < b

   1. Initial program 52.8%

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

      1. +-commutative 52.8%

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

      2. sqr-neg 52.8%

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

      3. unsub-neg 52.8%

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

      4. sqr-neg 52.8%

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

      5. sub-neg 52.8%

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

      6. +-commutative 52.8%

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

      7. *-commutative 52.8%

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

      8. associate-*r* 52.8%

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

      9. distribute-rgt-neg-in 52.8%

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

      10. fma-define 52.8%

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

      11. *-commutative 52.8%

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

      12. distribute-rgt-neg-in 52.8%

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

      13. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

      1. div-sub 52.4%

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

      2. *-un-lft-identity 52.4%

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

      3. *-commutative 52.4%

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

      4. times-frac 52.4%

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

      5. metadata-eval 52.4%

         \[\leadsto \color{blue}{0.5} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} - \frac{b}{a \cdot 2} \]

      6. pow2 52.4%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, \color{blue}{{b}^{2}}\right)}}{a} - \frac{b}{a \cdot 2} \]

      7. *-un-lft-identity 52.4%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - \frac{\color{blue}{1 \cdot b}}{a \cdot 2} \]

      8. *-commutative 52.4%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - \frac{1 \cdot b}{\color{blue}{2 \cdot a}} \]

      9. times-frac 52.4%

         \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - \color{blue}{\frac{1}{2} \cdot \frac{b}{a}} \]

      10. metadata-eval 52.4%

          \[\leadsto 0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - \color{blue}{0.5} \cdot \frac{b}{a} \]

   6. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}{a} - 0.5 \cdot \frac{b}{a}} \]

   7. Taylor expanded in a around 0 90.7%

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

      1. neg-mul-1 90.7%

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

      2. +-commutative 90.7%

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

      3. unsub-neg 90.7%

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

      4. mul-1-neg 90.7%

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

      5. unsub-neg 90.7%

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

      6. associate-*r/ 90.7%

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

   9. Simplified 90.7%

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

   10. Taylor expanded in c around inf 90.7%

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

       1. pow1 90.7%

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

       2. fma-neg 90.7%

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

       3. div-inv 90.7%

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

       4. pow-flip 90.7%

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

       5. metadata-eval 90.7%

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

       6. associate-/r* 90.7%

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

       7. pow-flip 90.7%

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

       8. metadata-eval 90.7%

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

   12. Applied egg-rr 90.7%

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

       1. unpow1 90.7%

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

       2. fma-neg 90.7%

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

       3. associate-*r* 90.7%

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

       4. *-commutative 90.7%

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

   14. Simplified 90.7%

       \[\leadsto a \cdot \color{blue}{\left({c}^{3} \cdot \left(\left(a \cdot -2\right) \cdot {b}^{-5} - \frac{{b}^{-3}}{c}\right)\right)} - \frac{c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left({c}^{3} \cdot \left(\left(a \cdot -2\right) \cdot {b}^{-5} - \frac{{b}^{-3}}{c}\right)\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.0)
   (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* 2.0 a))
   (/ (fma a (pow (/ c b) 2.0) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.0) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (2.0 * a);
	} else {
		tmp = fma(a, pow((c / b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.0)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-b));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 1

   1. Initial program 82.9%

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

      1. +-commutative 82.9%

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

      2. sqr-neg 82.9%

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

      3. unsub-neg 82.9%

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

      4. sqr-neg 82.9%

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

      5. sub-neg 82.9%

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

      6. +-commutative 82.9%

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

      7. *-commutative 82.9%

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

      8. associate-*r* 82.9%

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

      9. distribute-rgt-neg-in 82.9%

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

      10. fma-define 82.9%

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

      11. *-commutative 82.9%

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

      12. distribute-rgt-neg-in 82.9%

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

      13. metadata-eval 82.9%

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

   3. Simplified 82.9%

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

   if 1 < b

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in a around 0 85.2%

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

      1. mul-1-neg 85.2%

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

      2. unsub-neg 85.2%

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

      3. mul-1-neg 85.2%

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

      4. distribute-neg-frac2 85.2%

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

      5. associate-/l* 85.2%

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

   7. Simplified 85.2%

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

      1. expm1-log1p-u 77.8%

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

      2. distribute-frac-neg2 77.8%

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

      3. *-commutative 77.8%

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

      4. div-inv 77.8%

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

      5. pow-flip 77.8%

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

      6. metadata-eval 77.8%

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

   9. Applied egg-rr 77.8%

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

       1. expm1-undefine 58.1%

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

       2. sub-neg 58.1%

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

       3. log1p-undefine 58.1%

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

       4. rem-exp-log 65.4%

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

       5. sub-neg 65.4%

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

       6. distribute-neg-out 65.4%

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

       7. unsub-neg 65.4%

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

       8. *-commutative 65.4%

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

       9. metadata-eval 65.4%

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

   11. Simplified 65.4%

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

   12. Taylor expanded in c around 0 85.1%

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

   14. Simplified 85.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.02)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (/ (fma a (pow (/ c b) 2.0) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.02) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = fma(a, pow((c / b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.02)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.02], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.02

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. +-commutative 82.9%

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

      3. sqr-neg 82.9%

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

      4. unsub-neg 82.9%

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

      5. sqr-neg 82.9%

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

      6. fma-neg 83.1%

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

      7. distribute-lft-neg-in 83.1%

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

      8. *-commutative 83.1%

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

      9. *-commutative 83.1%

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

      10. distribute-rgt-neg-in 83.1%

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

      11. metadata-eval 83.1%

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

   3. Simplified 83.1%

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

   if 1.02 < b

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in a around 0 85.2%

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

      1. mul-1-neg 85.2%

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

      2. unsub-neg 85.2%

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

      3. mul-1-neg 85.2%

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

      4. distribute-neg-frac2 85.2%

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

      5. associate-/l* 85.2%

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

   7. Simplified 85.2%

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

      1. expm1-log1p-u 77.8%

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

      2. distribute-frac-neg2 77.8%

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

      3. *-commutative 77.8%

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

      4. div-inv 77.8%

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

      5. pow-flip 77.8%

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

      6. metadata-eval 77.8%

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

   9. Applied egg-rr 77.8%

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

       1. expm1-undefine 58.1%

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

       2. sub-neg 58.1%

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

       3. log1p-undefine 58.1%

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

       4. rem-exp-log 65.4%

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

       5. sub-neg 65.4%

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

       6. distribute-neg-out 65.4%

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

       7. unsub-neg 65.4%

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

       8. *-commutative 65.4%

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

       9. metadata-eval 65.4%

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

   11. Simplified 65.4%

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

   12. Taylor expanded in c around 0 85.1%

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

   14. Simplified 85.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.02)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (/ (fma a (pow (/ c b) 2.0) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.02) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = fma(a, pow((c / b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.02)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.02], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.02

   1. Initial program 82.9%

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

   if 1.02 < b

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in a around 0 85.2%

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

      1. mul-1-neg 85.2%

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

      2. unsub-neg 85.2%

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

      3. mul-1-neg 85.2%

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

      4. distribute-neg-frac2 85.2%

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

      5. associate-/l* 85.2%

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

   7. Simplified 85.2%

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

      1. expm1-log1p-u 77.8%

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

      2. distribute-frac-neg2 77.8%

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

      3. *-commutative 77.8%

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

      4. div-inv 77.8%

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

      5. pow-flip 77.8%

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

      6. metadata-eval 77.8%

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

   9. Applied egg-rr 77.8%

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

       1. expm1-undefine 58.1%

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

       2. sub-neg 58.1%

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

       3. log1p-undefine 58.1%

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

       4. rem-exp-log 65.4%

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

       5. sub-neg 65.4%

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

       6. distribute-neg-out 65.4%

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

       7. unsub-neg 65.4%

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

       8. *-commutative 65.4%

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

       9. metadata-eval 65.4%

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

   11. Simplified 65.4%

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

   12. Taylor expanded in c around 0 85.1%

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

   14. Simplified 85.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{c \cdot a}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.02)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (* c (- (/ -1.0 b) (/ (* c a) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.02) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = c * ((-1.0 / b) - ((c * a) / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.02d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = c * (((-1.0d0) / b) - ((c * a) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.02) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = c * ((-1.0 / b) - ((c * a) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.02:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = c * ((-1.0 / b) - ((c * a) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.02)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(c * a) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.02)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = c * ((-1.0 / b) - ((c * a) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.02], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.02

   1. Initial program 82.9%

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

   if 1.02 < b

   1. Initial program 52.8%

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

      1. *-commutative 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in c around 0 85.1%

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

      1. associate-*r/ 85.1%

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

      2. neg-mul-1 85.1%

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

      3. distribute-rgt-neg-in 85.1%

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

   7. Simplified 85.1%

      \[\leadsto \color{blue}{c \cdot \left(\frac{a \cdot \left(-c\right)}{{b}^{3}} - \frac{1}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \frac{c \cdot a}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

   1. *-commutative 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in a around 0 81.5%

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

   1. mul-1-neg 81.5%

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

   2. unsub-neg 81.5%

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

   3. mul-1-neg 81.5%

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

   4. distribute-neg-frac2 81.5%

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

   5. associate-/l* 81.5%

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

7. Simplified 81.5%

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

8. Taylor expanded in a around inf 81.3%

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

   1. associate-*r/ 81.3%

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

   2. *-commutative 81.3%

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

   3. associate-/r* 81.3%

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

   4. mul-1-neg 81.3%

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

10. Simplified 81.3%

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

11. Taylor expanded in b around inf 81.3%

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

    1. mul-1-neg 81.3%

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

    2. unsub-neg 81.3%

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

    3. associate-*r/ 81.3%

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

    4. neg-mul-1 81.3%

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

    5. unpow2 81.3%

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

    6. unpow2 81.3%

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

    7. times-frac 81.3%

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

    8. unpow2 81.3%

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

13. Simplified 81.3%

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

14. Final simplification 81.3%

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

Alternative 12: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

   1. *-commutative 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in c around 0 81.4%

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

   1. associate-*r/ 81.4%

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

   2. neg-mul-1 81.4%

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

   3. distribute-rgt-neg-in 81.4%

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

7. Simplified 81.4%

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

8. Final simplification 81.4%

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

Alternative 13: 64.5% 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 57.2%

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

   1. *-commutative 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in b around inf 62.6%

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

   1. associate-*r/ 62.6%

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

   2. mul-1-neg 62.6%

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

7. Simplified 62.6%

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

8. Final simplification 62.6%

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

Alternative 14: 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 57.2%

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

   1. *-commutative 57.2%

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

3. Simplified 57.2%

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

5. Taylor expanded in a around 0 81.5%

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

   1. mul-1-neg 81.5%

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

   2. unsub-neg 81.5%

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

   3. mul-1-neg 81.5%

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

   4. distribute-neg-frac2 81.5%

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

   5. associate-/l* 81.5%

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

7. Simplified 81.5%

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

   1. expm1-log1p-u 74.8%

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

   2. distribute-frac-neg2 74.8%

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

   3. *-commutative 74.8%

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

   4. div-inv 74.8%

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

   5. pow-flip 74.8%

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

   6. metadata-eval 74.8%

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

9. Applied egg-rr 74.8%

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

    1. expm1-undefine 57.7%

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

    2. sub-neg 57.7%

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

    3. log1p-undefine 57.7%

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

    4. rem-exp-log 64.3%

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

    5. sub-neg 64.3%

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

    6. distribute-neg-out 64.3%

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

    7. unsub-neg 64.3%

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

    8. *-commutative 64.3%

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

    9. metadata-eval 64.3%

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

11. Simplified 64.3%

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

12. Taylor expanded in c around 0 52.4%

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

13. Taylor expanded in c around 0 3.2%

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

14. Final simplification 3.2%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))