Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.6%

Time: 13.0s

Alternatives: 9

Speedup: 3.6×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.6%

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

Alternative 2: 97.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 97.6%

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

7. Applied rewrites 97.6%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)}{\left(a \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)\right)\right) \cdot 0.05}, -0.25, \frac{-2 \cdot \frac{\left(a \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} - c \cdot \left(c \cdot a\right)}{b \cdot b} - c\right)}{b} \]

8. Taylor expanded in c around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)}{\left(a \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)\right)\right) \cdot \frac{1}{20}}, \frac{-1}{4}, \frac{{c}^{2} \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} - a\right)}{b \cdot b} - c\right)}{b} \]
9. Step-by-step derivation

10. Applied rewrites 97.6%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)}{\left(a \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b\right)\right)\right) \cdot 0.05}, -0.25, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c\right)}{b} \]
11. Step-by-step derivation

12. Applied rewrites 97.6%

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

13. Final simplification 97.6%

    \[\leadsto \frac{\mathsf{fma}\left(\left(c \cdot \left(c \cdot \left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\right)\right) \cdot \frac{c \cdot c}{\left(b \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, -5, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c\right)}{b} \]
14. Add Preprocessing

Alternative 3: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 96.9%

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

Alternative 4: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 96.9%

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

7. Applied rewrites 96.9%

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

8. Final simplification 96.9%

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

Alternative 5: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{1}{b}\right), \frac{b}{-c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma a (fma (* a -2.0) (* (/ c (* b (* b b))) -0.5) (/ 1.0 b)) (/ b (- c)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(a, fma((a * -2.0), ((c / (b * (b * b))) * -0.5), (1.0 / b)), (b / -c));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(a, fma(Float64(a * -2.0), Float64(Float64(c / Float64(b * Float64(b * b))) * -0.5), Float64(1.0 / b)), Float64(b / Float64(-c))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(a * N[(N[(a * -2.0), $MachinePrecision] * N[(N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[(b / (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 17.4

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 17.4

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

4. Applied rewrites 17.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 17.4

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

   9. lift-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 17.4

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 17.4

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

6. Applied rewrites 17.4%

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

8. Applied rewrites 17.4%

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

9. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. associate-*r* N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. distribute-rgt-out N/A

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

    7. metadata-eval N/A

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

    8. lower-*.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. cube-mult N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. unpow2 N/A

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

    14. lower-*.f64 N/A

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

    15. lower-/.f64 N/A

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

11. Applied rewrites 96.7%

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

12. Final simplification 96.7%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{1}{b}\right), \frac{b}{-c}\right)} \]
13. Add Preprocessing

Alternative 6: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.6

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

5. Applied rewrites 95.6%

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

6. Final simplification 95.6%

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

Alternative 7: 95.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 17.4

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 17.4

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

4. Applied rewrites 17.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 17.4

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

   9. lift-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 17.4

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 17.4

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

6. Applied rewrites 17.4%

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

8. Applied rewrites 17.4%

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

9. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

    4. lower--.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 95.4

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

11. Applied rewrites 95.4%

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

Alternative 8: 90.3% accurate, 3.6× 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 17.4%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   4. lower-neg.f64 90.9

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

5. Applied rewrites 90.9%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Alternative 9: 3.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 17.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 17.4

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. distribute-rgt-neg-in N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 17.4

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

4. Applied rewrites 17.4%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

6. Applied rewrites 17.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 18.9

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

8. Applied rewrites 18.9%

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

9. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{-2 \cdot b + 2 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

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

    2. distribute-rgt-out N/A

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

    3. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(b \cdot \color{blue}{0}\right)}{a} \]

    4. mul0-rgt N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{0}}{a} \]

    5. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{0}}{a} \]

    6. lower-/.f64 3.3

       \[\leadsto \color{blue}{\frac{0}{a}} \]

11. Applied rewrites 3.3%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))