Quadratic roots, wide range

Percentage Accurate: 17.8% → 99.9%

Time: 24.7s

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: 17.8% 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: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{c \cdot a}\\ 4 \cdot \frac{c}{2 \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(2, t\_0, b\right) \cdot \mathsf{fma}\left(-2, t\_0, b\right)}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* c a))))
   (*
    4.0
    (/ c (* 2.0 (- (- b) (sqrt (* (fma 2.0 t_0 b) (fma -2.0 t_0 b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((c * a));
	return 4.0 * (c / (2.0 * (-b - sqrt((fma(2.0, t_0, b) * fma(-2.0, t_0, b))))));
}

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(c * a))
	return Float64(4.0 * Float64(c / Float64(2.0 * Float64(Float64(-b) - sqrt(Float64(fma(2.0, t_0, b) * fma(-2.0, t_0, b)))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(c * a), $MachinePrecision]], $MachinePrecision]}, N[(4.0 * N[(c / N[(2.0 * N[((-b) - N[Sqrt[N[(N[(2.0 * t$95$0 + b), $MachinePrecision] * N[(-2.0 * t$95$0 + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 14.6%

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

   1. *-commutative 14.6%

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

3. Simplified 14.6%

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

   1. add-sqr-sqrt 14.6%

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

   2. difference-of-squares 14.8%

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

   3. associate-*l* 14.8%

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

   4. sqrt-prod 14.8%

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

   5. metadata-eval 14.8%

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

   6. associate-*l* 14.8%

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

   7. sqrt-prod 14.8%

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

   8. metadata-eval 14.8%

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

6. Applied egg-rr 14.8%

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

   1. *-commutative 14.8%

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

   2. cancel-sign-sub-inv 14.8%

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

   3. metadata-eval 14.8%

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

8. Simplified 14.8%

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

   1. flip-+ 14.8%

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

   2. pow2 14.8%

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

   3. add-sqr-sqrt 15.2%

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

   4. +-commutative 15.2%

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

   5. *-commutative 15.2%

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

   6. fma-define 15.2%

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

   7. +-commutative 15.2%

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

   8. fma-define 15.2%

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

10. Applied egg-rr 15.2%

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

    1. unpow2 15.2%

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

    2. sqr-neg 15.2%

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

    3. unpow2 15.2%

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

    4. fma-undefine 15.2%

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

    5. *-commutative 15.2%

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

    6. fma-define 15.2%

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

    7. fma-undefine 15.2%

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

    8. *-commutative 15.2%

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

    9. fma-define 15.2%

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

12. Simplified 15.2%

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

13. Taylor expanded in b around 0 99.5%

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

    1. div-inv 99.3%

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

15. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. times-frac 99.4%

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

    3. *-lft-identity 99.4%

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

    4. associate-/l* 99.4%

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

    5. associate-*l* 99.4%

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

    6. times-frac 99.9%

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

    7. *-inverses 99.9%

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

17. Simplified 99.9%

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

18. Final simplification 99.9%

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

Alternative 2: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (- (/ c (- b)) (* a (/ (pow c 2.0) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (c / -b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c / -b) - (a * ((c ** 2.0d0) / (b ** 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 14.6%

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

   1. *-commutative 14.6%

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

3. Simplified 14.6%

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

5. Taylor expanded in b around inf 95.1%

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

6. Taylor expanded in a around 0 95.5%

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

   1. mul-1-neg 95.5%

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

   2. unsub-neg 95.5%

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

   3. associate-*r/ 95.5%

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

   4. mul-1-neg 95.5%

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

   5. associate-/l* 95.5%

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

8. Simplified 95.5%

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

9. Final simplification 95.5%

   \[\leadsto \frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \]
10. Add Preprocessing

Alternative 3: 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(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 14.6%

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

   1. *-commutative 14.6%

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

3. Simplified 14.6%

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

5. Taylor expanded in b around inf 92.3%

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

   1. mul-1-neg 92.3%

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

   2. distribute-neg-frac 92.3%

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

7. Simplified 92.3%

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

8. Final simplification 92.3%

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

Alternative 4: 3.3% accurate, 38.7× 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 14.6%

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

   1. *-commutative 14.6%

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

3. Simplified 14.6%

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

   1. add-sqr-sqrt 14.6%

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

   2. difference-of-squares 14.8%

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

   3. associate-*l* 14.8%

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

   4. sqrt-prod 14.8%

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

   5. metadata-eval 14.8%

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

   6. associate-*l* 14.8%

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

   7. sqrt-prod 14.8%

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

   8. metadata-eval 14.8%

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

6. Applied egg-rr 14.8%

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

   1. *-commutative 14.8%

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

   2. cancel-sign-sub-inv 14.8%

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

   3. metadata-eval 14.8%

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

8. Simplified 14.8%

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

9. Taylor expanded in b around inf 3.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 3.3%

       \[\leadsto \color{blue}{\frac{0.25 \cdot \left(-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}\right)}{a}} \]

    2. distribute-rgt-out 3.3%

       \[\leadsto \frac{0.25 \cdot \color{blue}{\left(\sqrt{a \cdot c} \cdot \left(-2 + 2\right)\right)}}{a} \]

    3. metadata-eval 3.3%

       \[\leadsto \frac{0.25 \cdot \left(\sqrt{a \cdot c} \cdot \color{blue}{0}\right)}{a} \]

    4. mul0-rgt 3.3%

       \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 3.3%

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

11. Simplified 3.3%

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

12. Final simplification 3.3%

    \[\leadsto \frac{0}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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)))