Quadratic roots, wide range

Percentage Accurate: 17.8% → 99.7%

Time: 9.3s

Alternatives: 5

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: 17.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.0%

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

   1. flip-+ 16.0%

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

   2. pow2 16.0%

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

   3. add-sqr-sqrt 16.5%

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

   4. *-commutative 16.5%

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

   5. *-commutative 16.5%

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

   6. *-commutative 16.5%

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

   7. *-commutative 16.5%

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

3. Applied egg-rr 16.5%

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

   1. unpow2 16.5%

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

   2. sqr-neg 16.5%

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

   3. sub-neg 16.5%

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

   4. +-commutative 16.5%

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

   5. distribute-rgt-neg-in 16.5%

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

   6. fma-def 16.5%

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

   7. distribute-rgt-neg-in 16.5%

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

   8. metadata-eval 16.5%

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

   9. sub-neg 16.5%

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

   10. +-commutative 16.5%

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

   11. distribute-rgt-neg-in 16.5%

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

   12. fma-def 16.5%

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

   13. distribute-rgt-neg-in 16.5%

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

   14. metadata-eval 16.5%

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

5. Simplified 16.5%

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

6. Taylor expanded in b around 0 99.5%

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

   1. associate-*r* 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

8. Simplified 99.5%

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

   1. div-inv 99.3%

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

   2. *-commutative 99.3%

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

10. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. times-frac 99.4%

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

    3. associate-*l* 99.4%

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

    4. *-lft-identity 99.4%

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

    5. times-frac 99.5%

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

    6. *-commutative 99.5%

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

12. Simplified 99.5%

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

    1. associate-*r/ 99.8%

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

    2. *-commutative 99.8%

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

14. Applied egg-rr 99.8%

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

15. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / a) * ((a * 4.0d0) / (2.0d0 * (-b - sqrt(((b * b) + (c * (a * (-4.0d0))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.0%

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

   1. flip-+ 16.0%

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

   2. pow2 16.0%

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

   3. add-sqr-sqrt 16.5%

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

   4. *-commutative 16.5%

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

   5. *-commutative 16.5%

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

   6. *-commutative 16.5%

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

   7. *-commutative 16.5%

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

3. Applied egg-rr 16.5%

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

   1. unpow2 16.5%

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

   2. sqr-neg 16.5%

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

   3. sub-neg 16.5%

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

   4. +-commutative 16.5%

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

   5. distribute-rgt-neg-in 16.5%

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

   6. fma-def 16.5%

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

   7. distribute-rgt-neg-in 16.5%

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

   8. metadata-eval 16.5%

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

   9. sub-neg 16.5%

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

   10. +-commutative 16.5%

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

   11. distribute-rgt-neg-in 16.5%

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

   12. fma-def 16.5%

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

   13. distribute-rgt-neg-in 16.5%

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

   14. metadata-eval 16.5%

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

5. Simplified 16.5%

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

6. Taylor expanded in b around 0 99.5%

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

   1. associate-*r* 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

8. Simplified 99.5%

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

   1. div-inv 99.3%

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

   2. *-commutative 99.3%

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

10. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. times-frac 99.4%

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

    3. associate-*l* 99.4%

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

    4. *-lft-identity 99.4%

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

    5. times-frac 99.5%

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

    6. *-commutative 99.5%

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

12. Simplified 99.5%

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

    1. fma-udef 99.5%

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

14. Applied egg-rr 99.5%

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

15. Final simplification 99.5%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.0%

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

   1. flip-+ 16.0%

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

   2. pow2 16.0%

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

   3. add-sqr-sqrt 16.5%

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

   4. *-commutative 16.5%

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

   5. *-commutative 16.5%

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

   6. *-commutative 16.5%

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

   7. *-commutative 16.5%

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

3. Applied egg-rr 16.5%

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

   1. unpow2 16.5%

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

   2. sqr-neg 16.5%

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

   3. sub-neg 16.5%

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

   4. +-commutative 16.5%

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

   5. distribute-rgt-neg-in 16.5%

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

   6. fma-def 16.5%

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

   7. distribute-rgt-neg-in 16.5%

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

   8. metadata-eval 16.5%

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

   9. sub-neg 16.5%

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

   10. +-commutative 16.5%

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

   11. distribute-rgt-neg-in 16.5%

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

   12. fma-def 16.5%

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

   13. distribute-rgt-neg-in 16.5%

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

   14. metadata-eval 16.5%

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

5. Simplified 16.5%

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

6. Taylor expanded in b around 0 99.5%

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

   1. associate-*r* 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

8. Simplified 99.5%

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

   1. div-inv 99.3%

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

10. Applied egg-rr 99.3%

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

    1. associate-*r/ 99.5%

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

    2. *-rgt-identity 99.5%

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

    3. associate-/l* 99.4%

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

    4. associate-/r/ 99.7%

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

    5. *-commutative 99.7%

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

12. Simplified 99.7%

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

    1. fma-udef 99.5%

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

14. Applied egg-rr 99.7%

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

15. Final simplification 99.7%

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

Alternative 4: 95.1% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.0%

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

   1. neg-sub0 16.0%

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

   2. associate-+l- 16.0%

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

   3. sub0-neg 16.0%

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

   4. neg-mul-1 16.0%

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

   5. associate-*l/ 16.0%

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

   6. *-commutative 16.0%

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

   7. associate-/r* 16.0%

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

   8. /-rgt-identity 16.0%

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

   9. metadata-eval 16.0%

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

3. Simplified 16.0%

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

4. Taylor expanded in b around inf 95.9%

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

   1. distribute-lft-out 95.9%

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

   2. mul-1-neg 95.9%

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

   3. +-commutative 95.9%

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

   4. unpow2 95.9%

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

   5. associate-*l* 95.9%

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

6. Simplified 95.9%

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

   1. unpow3 95.9%

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

8. Applied egg-rr 95.9%

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

9. Final simplification 95.9%

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

Alternative 5: 90.4% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(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 16.0%

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

   1. neg-sub0 16.0%

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

   2. associate-+l- 16.0%

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

   3. sub0-neg 16.0%

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

   4. neg-mul-1 16.0%

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

   5. associate-*l/ 16.0%

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

   6. *-commutative 16.0%

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

   7. associate-/r* 16.0%

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

   8. /-rgt-identity 16.0%

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

   9. metadata-eval 16.0%

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

3. Simplified 16.0%

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

4. Taylor expanded in b around inf 92.0%

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

   1. associate-*r/ 92.0%

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

   2. neg-mul-1 92.0%

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

6. Simplified 92.0%

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

7. Final simplification 92.0%

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

Reproduce

herbie shell --seed 2023228 
(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)))