Quadratic roots, wide range

Percentage Accurate: 17.6% → 99.7%

Time: 8.8s

Alternatives: 4

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.1%

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

   1. *-un-lft-identity 15.1%

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

   2. prod-diff 15.2%

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

3. Applied egg-rr 15.1%

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

   1. +-commutative 15.1%

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

   2. fma-udef 15.1%

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

   3. *-rgt-identity 15.1%

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

   4. *-rgt-identity 15.1%

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

   5. distribute-rgt-out 15.1%

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

   6. *-rgt-identity 15.1%

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

5. Simplified 15.1%

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

6. Taylor expanded in b around 0 15.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 15.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)} \cdot \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}}{2 \cdot a} \]

   2. add-sqr-sqrt 15.3%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \color{blue}{\left(\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)\right)}}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   3. associate--l+ 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \color{blue}{\left({b}^{2} + \left(-8 \cdot \left(c \cdot a\right) - -4 \cdot \left(c \cdot a\right)\right)\right)}}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   4. unpow2 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(\color{blue}{b \cdot b} + \left(-8 \cdot \left(c \cdot a\right) - -4 \cdot \left(c \cdot a\right)\right)\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   5. distribute-rgt-out-- 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + \color{blue}{\left(c \cdot a\right) \cdot \left(-8 - -4\right)}\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   6. metadata-eval 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + \left(c \cdot a\right) \cdot \color{blue}{-4}\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   7. *-commutative 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + \color{blue}{\left(a \cdot c\right)} \cdot -4\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   8. associate-*r* 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + \color{blue}{a \cdot \left(c \cdot -4\right)}\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   9. associate--l+ 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-8 \cdot \left(c \cdot a\right) - -4 \cdot \left(c \cdot a\right)\right)}}}}{2 \cdot a} \]

   10. unpow2 15.4%

       \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} + \left(-8 \cdot \left(c \cdot a\right) - -4 \cdot \left(c \cdot a\right)\right)}}}{2 \cdot a} \]

8. Applied egg-rr 15.4%

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

   1. sqr-neg 15.4%

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

   2. unpow2 15.4%

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

   3. unpow2 15.4%

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

   4. associate--r+ 99.4%

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

   5. +-inverses 99.4%

      \[\leadsto \frac{\frac{\color{blue}{0} - a \cdot \left(c \cdot -4\right)}{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}{2 \cdot a} \]

   6. neg-sub0 99.4%

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

   7. associate-*r* 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. *-commutative 99.4%

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

   11. *-commutative 99.4%

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

   12. unpow2 99.4%

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

   13. +-commutative 99.4%

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

   14. associate-*r* 99.4%

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

10. Simplified 99.4%

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

    1. div-inv 99.2%

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

    2. *-commutative 99.2%

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

12. Applied egg-rr 99.2%

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

    1. associate-*l/ 99.4%

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

    2. associate-*r/ 99.7%

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

    3. *-rgt-identity 99.7%

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

    4. *-commutative 99.7%

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

    5. times-frac 99.7%

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

    6. metadata-eval 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.1%

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

   1. *-un-lft-identity 15.1%

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

   2. prod-diff 15.2%

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

3. Applied egg-rr 15.1%

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

   1. +-commutative 15.1%

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

   2. fma-udef 15.1%

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

   3. *-rgt-identity 15.1%

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

   4. *-rgt-identity 15.1%

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

   5. distribute-rgt-out 15.1%

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

   6. *-rgt-identity 15.1%

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

5. Simplified 15.1%

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

6. Taylor expanded in b around 0 15.0%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 15.0%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)} \cdot \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}}{2 \cdot a} \]

   2. add-sqr-sqrt 15.3%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \color{blue}{\left(\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)\right)}}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   3. associate--l+ 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \color{blue}{\left({b}^{2} + \left(-8 \cdot \left(c \cdot a\right) - -4 \cdot \left(c \cdot a\right)\right)\right)}}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   4. unpow2 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(\color{blue}{b \cdot b} + \left(-8 \cdot \left(c \cdot a\right) - -4 \cdot \left(c \cdot a\right)\right)\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   5. distribute-rgt-out-- 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + \color{blue}{\left(c \cdot a\right) \cdot \left(-8 - -4\right)}\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   6. metadata-eval 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + \left(c \cdot a\right) \cdot \color{blue}{-4}\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   7. *-commutative 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + \color{blue}{\left(a \cdot c\right)} \cdot -4\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   8. associate-*r* 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + \color{blue}{a \cdot \left(c \cdot -4\right)}\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + -8 \cdot \left(c \cdot a\right)\right) - -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   9. associate--l+ 15.4%

      \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + \left(-8 \cdot \left(c \cdot a\right) - -4 \cdot \left(c \cdot a\right)\right)}}}}{2 \cdot a} \]

   10. unpow2 15.4%

       \[\leadsto \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \left(b \cdot b + a \cdot \left(c \cdot -4\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} + \left(-8 \cdot \left(c \cdot a\right) - -4 \cdot \left(c \cdot a\right)\right)}}}{2 \cdot a} \]

8. Applied egg-rr 15.4%

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

9. Taylor expanded in b around 0 99.4%

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

    1. *-commutative 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.1%

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

   1. /-rgt-identity 15.1%

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

   2. metadata-eval 15.1%

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

   3. associate-/l* 15.1%

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

   4. associate-*r/ 15.1%

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

   5. +-commutative 15.1%

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

   6. unsub-neg 15.1%

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

   7. fma-neg 15.2%

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

   8. associate-*l* 15.2%

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

   9. *-commutative 15.2%

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

   10. distribute-rgt-neg-in 15.2%

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

   11. metadata-eval 15.2%

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

   12. associate-/r* 15.2%

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

   13. metadata-eval 15.2%

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

   14. metadata-eval 15.2%

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

3. Simplified 15.2%

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

4. Taylor expanded in b around inf 96.0%

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

   1. +-commutative 96.0%

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

   2. mul-1-neg 96.0%

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

   3. unsub-neg 96.0%

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

   4. mul-1-neg 96.0%

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

   5. associate-/l* 96.0%

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

   6. unpow2 96.0%

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

6. Simplified 96.0%

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

7. Final simplification 96.0%

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

Alternative 4: 90.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(-c) / 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 15.1%

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

   1. /-rgt-identity 15.1%

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

   2. metadata-eval 15.1%

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

   3. associate-/l* 15.1%

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

   4. associate-*r/ 15.1%

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

   5. +-commutative 15.1%

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

   6. unsub-neg 15.1%

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

   7. fma-neg 15.2%

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

   8. associate-*l* 15.2%

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

   9. *-commutative 15.2%

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

   10. distribute-rgt-neg-in 15.2%

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

   11. metadata-eval 15.2%

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

   12. associate-/r* 15.2%

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

   13. metadata-eval 15.2%

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

   14. metadata-eval 15.2%

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

3. Simplified 15.2%

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

4. Taylor expanded in b around inf 92.4%

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

   1. mul-1-neg 92.4%

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

6. Simplified 92.4%

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

7. Final simplification 92.4%

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

Reproduce

herbie shell --seed 2023187 
(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)))