Quadratic roots, wide range

Percentage Accurate: 17.9% → 99.7%

Time: 12.1s

Alternatives: 6

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: 17.9% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

   1. *-commutative 19.4%

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

3. Simplified 19.4%

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

   1. add-sqr-sqrt 19.4%

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

   2. difference-of-squares 19.4%

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

   3. associate-*l* 19.4%

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

   4. sqrt-prod 19.4%

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

   5. metadata-eval 19.4%

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

   6. associate-*l* 19.4%

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

   7. sqrt-prod 19.4%

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

   8. metadata-eval 19.4%

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

5. Applied egg-rr 19.4%

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

   1. *-commutative 19.4%

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

   2. cancel-sign-sub-inv 19.4%

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

   3. metadata-eval 19.4%

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

7. Simplified 19.4%

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

   1. flip-+ 19.2%

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

   2. pow2 19.2%

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

   3. add-sqr-sqrt 19.6%

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

   4. +-commutative 19.6%

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

   5. *-commutative 19.6%

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

   6. fma-def 19.6%

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

   7. +-commutative 19.6%

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

   8. fma-def 19.6%

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

9. Applied egg-rr 19.6%

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

    1. unpow2 19.6%

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

    2. sqr-neg 19.6%

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

    3. unpow2 19.6%

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

    4. fma-udef 19.6%

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

    5. *-commutative 19.6%

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

    6. fma-def 19.6%

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

    7. fma-udef 19.6%

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

    8. *-commutative 19.6%

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

    9. fma-def 19.6%

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

11. Simplified 19.6%

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

12. Taylor expanded in b around 0 99.4%

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

    1. associate-*r* 99.4%

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

14. Simplified 99.4%

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

    1. expm1-log1p-u 82.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a \cdot c}, 2, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}}}{a \cdot 2}\right)\right)} \]

    2. expm1-udef 22.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a \cdot c}, 2, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}}}{a \cdot 2}\right)} - 1} \]

    3. associate-/l/ 22.9%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(4 \cdot a\right) \cdot c}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a \cdot c}, 2, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}\right)}}\right)} - 1 \]

    4. associate-*l* 22.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a \cdot c}, 2, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}\right)}\right)} - 1 \]

16. Applied egg-rr 22.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4 \cdot \left(a \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a \cdot c}, 2, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}\right)}\right)} - 1} \]
17. Step-by-step derivation

    1. expm1-def 82.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4 \cdot \left(a \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(\sqrt{a \cdot c}, 2, b\right) \cdot \mathsf{fma}\left(-2, \sqrt{a \cdot c}, b\right)}\right)}\right)\right)} \]

    2. expm1-log1p 99.3%

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

    3. associate-/r* 99.7%

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

    4. *-commutative 99.7%

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

    5. times-frac 99.7%

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

    6. metadata-eval 99.7%

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

18. Simplified 99.7%

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

19. Final simplification 99.7%

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

Alternative 2: 96.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

   1. *-commutative 19.4%

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

3. Simplified 19.4%

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

4. Taylor expanded in b around inf 96.7%

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

   1. associate-+r+ 96.7%

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

   2. mul-1-neg 96.7%

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

   3. unsub-neg 96.7%

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

   4. mul-1-neg 96.7%

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

   5. unsub-neg 96.7%

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

   6. associate-*r/ 96.7%

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

   7. *-commutative 96.7%

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

   8. associate-/l* 96.7%

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

6. Simplified 96.7%

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

7. Final simplification 96.7%

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

Alternative 3: 95.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-c / b) - (a / (pow(b, 3.0) / pow(c, 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 = (-c / b) - (a / ((b ** 3.0d0) / (c ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

   1. *-commutative 19.4%

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

3. Simplified 19.4%

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

4. Taylor expanded in b around inf 95.1%

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

   1. mul-1-neg 95.1%

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

   2. unsub-neg 95.1%

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

   3. mul-1-neg 95.1%

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

   4. distribute-neg-frac 95.1%

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

   5. associate-/l* 95.1%

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

6. Simplified 95.1%

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

7. Final simplification 95.1%

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

Alternative 4: 95.0% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.4%

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

   1. *-commutative 19.4%

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

3. Simplified 19.4%

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

4. Taylor expanded in b around inf 94.7%

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

   1. distribute-lft-out 94.7%

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

   2. associate-/l* 94.6%

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

   3. associate-/l* 94.6%

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

6. Simplified 94.6%

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

   1. clear-num 94.7%

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

   2. inv-pow 94.7%

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

   3. +-commutative 94.7%

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

   4. associate-/r/ 94.7%

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

   5. fma-def 94.7%

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

   6. associate-/r/ 94.5%

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

8. Applied egg-rr 94.5%

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

   1. unpow-1 94.5%

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

   2. times-frac 94.5%

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

   3. *-commutative 94.5%

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

10. Simplified 94.5%

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

11. Taylor expanded in a around 0 94.9%

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

    1. +-commutative 94.9%

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

    2. neg-mul-1 94.9%

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

    3. unsub-neg 94.9%

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

13. Simplified 94.9%

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

14. Final simplification 94.9%

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

Alternative 5: 90.3% 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 19.4%

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

   1. *-commutative 19.4%

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

3. Simplified 19.4%

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

4. Taylor expanded in b around inf 89.7%

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

   1. mul-1-neg 89.7%

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

   2. distribute-neg-frac 89.7%

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

6. Simplified 89.7%

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

7. Final simplification 89.7%

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

Alternative 6: 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 19.4%

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

   1. *-commutative 19.4%

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

3. Simplified 19.4%

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

   1. add-sqr-sqrt 19.4%

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

   2. difference-of-squares 19.4%

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

   3. associate-*l* 19.4%

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

   4. sqrt-prod 19.4%

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

   5. metadata-eval 19.4%

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

   6. associate-*l* 19.4%

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

   7. sqrt-prod 19.4%

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

   8. metadata-eval 19.4%

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

5. Applied egg-rr 19.4%

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

   1. *-commutative 19.4%

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

   2. cancel-sign-sub-inv 19.4%

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

   3. metadata-eval 19.4%

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

7. Simplified 19.4%

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

8. Taylor expanded in b around inf 3.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
9. 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%

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

10. Simplified 3.3%

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

11. Final simplification 3.3%

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

Reproduce

herbie shell --seed 2023306 
(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)))