Quadratic roots, wide range

Percentage Accurate: 18.4% → 97.5%

Time: 11.5s

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: 18.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (* -0.25 (* (/ (pow (* a c) 4.0) a) (/ 20.0 (pow b 7.0))))
    (/ (* a (pow c 2.0)) (pow b 3.0)))
   (/ c b))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * ((pow((a * c), 4.0) / a) * (20.0 / pow(b, 7.0)))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * ((((a * c) ** 4.0d0) / a) * (20.0d0 / (b ** 7.0d0)))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * ((Math.pow((a * c), 4.0) / a) * (20.0 / Math.pow(b, 7.0)))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) - (c / b));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * ((math.pow((a * c), 4.0) / a) * (20.0 / math.pow(b, 7.0)))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) - (c / b))

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * ((((a * c) ^ 4.0) / a) * (20.0 / (b ^ 7.0)))) - ((a * (c ^ 2.0)) / (b ^ 3.0))) - (c / b));
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

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

5. Taylor expanded in b around inf 97.9%

   \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 97.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -2\right)}}^{2}}{a \cdot {b}^{7}}\right)\right) \]

   2. unpow-prod-down 97.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left({a}^{2} \cdot {c}^{2}\right)}^{2} \cdot {-2}^{2}}}{a \cdot {b}^{7}}\right)\right) \]

   3. pow-prod-down 97.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\color{blue}{\left({\left(a \cdot c\right)}^{2}\right)}}^{2} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

   4. pow-pow 97.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

   5. metadata-eval 97.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{\color{blue}{4}} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

   6. metadata-eval 97.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{4} \cdot \color{blue}{4}}{a \cdot {b}^{7}}\right)\right) \]

7. Applied egg-rr 97.9%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{4} \cdot 4}}{a \cdot {b}^{7}}\right)\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 97.9%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({a}^{4} \cdot {c}^{4}\right)\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   2. expm1-udef 97.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({a}^{4} \cdot {c}^{4}\right)} - 1\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   3. pow-prod-down 97.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(a \cdot c\right)}^{4}}\right)} - 1\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

9. Applied egg-rr 97.4%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(a \cdot c\right)}^{4}\right)} - 1\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]
10. Step-by-step derivation

    1. expm1-def 97.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(a \cdot c\right)}^{4}\right)\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

    2. expm1-log1p 97.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{{\left(a \cdot c\right)}^{4}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

11. Simplified 97.9%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{{\left(a \cdot c\right)}^{4}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

12. Taylor expanded in c around 0 97.9%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
13. Step-by-step derivation

    1. distribute-rgt-out 97.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

    2. associate-*r* 97.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}\right)\right) \]

    3. *-commutative 97.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}\right)\right) \]

    4. times-frac 97.9%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{4 + 16}{{b}^{7}}\right)}\right)\right) \]

14. Simplified 97.9%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right)}\right)\right) \]

15. Final simplification 97.9%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \]
16. Add Preprocessing

Alternative 2: 96.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

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

5. Taylor expanded in b around inf 96.9%

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

   1. associate-+r+ 96.9%

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

   2. mul-1-neg 96.9%

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

   3. unsub-neg 96.9%

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

   4. mul-1-neg 96.9%

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

   5. unsub-neg 96.9%

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

   6. associate-*r/ 96.9%

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

   7. associate-/l* 96.9%

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

   8. *-commutative 96.9%

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

   9. associate-/l* 96.9%

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

   10. associate-/r/ 96.9%

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

7. Simplified 96.9%

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

8. Final simplification 96.9%

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

Alternative 3: 95.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

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

5. Taylor expanded in b around inf 95.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
6. 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. associate-/r/ 95.1%

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

7. Simplified 95.1%

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

8. Final simplification 95.1%

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

Alternative 4: 94.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1} \end{array} \]

FPCore

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

C

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

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 / b) - (b / c)) ** (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

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

5. Taylor expanded in b around inf 94.7%

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

   1. add-sqr-sqrt 94.7%

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

   2. pow2 94.7%

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

   3. sqrt-div 94.7%

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

   4. pow-prod-down 94.7%

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

   5. sqrt-pow1 94.7%

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

   6. metadata-eval 94.7%

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

   7. pow1 94.7%

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

   8. sqrt-pow1 94.7%

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

   9. metadata-eval 94.7%

      \[\leadsto \frac{-2 \cdot \frac{a \cdot c}{b} + -2 \cdot {\left(\frac{a \cdot c}{{b}^{\color{blue}{1.5}}}\right)}^{2}}{a \cdot 2} \]

7. Applied egg-rr 94.7%

   \[\leadsto \frac{-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \color{blue}{{\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}}{a \cdot 2} \]
8. Step-by-step derivation

   1. clear-num 94.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{-2 \cdot \frac{a \cdot c}{b} + -2 \cdot {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}}} \]

   2. inv-pow 94.6%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{-2 \cdot \frac{a \cdot c}{b} + -2 \cdot {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}\right)}^{-1}} \]

   3. *-commutative 94.6%

      \[\leadsto {\left(\frac{\color{blue}{2 \cdot a}}{-2 \cdot \frac{a \cdot c}{b} + -2 \cdot {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}\right)}^{-1} \]

   4. distribute-lft-out 94.6%

      \[\leadsto {\left(\frac{2 \cdot a}{\color{blue}{-2 \cdot \left(\frac{a \cdot c}{b} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}\right)}}\right)}^{-1} \]

   5. times-frac 94.6%

      \[\leadsto {\color{blue}{\left(\frac{2}{-2} \cdot \frac{a}{\frac{a \cdot c}{b} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}\right)}}^{-1} \]

   6. metadata-eval 94.6%

      \[\leadsto {\left(\color{blue}{-1} \cdot \frac{a}{\frac{a \cdot c}{b} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}\right)}^{-1} \]

   7. associate-/l* 94.7%

      \[\leadsto {\left(-1 \cdot \frac{a}{\color{blue}{\frac{a}{\frac{b}{c}}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}\right)}^{-1} \]

   8. unpow2 94.7%

      \[\leadsto {\left(-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + \color{blue}{\frac{a \cdot c}{{b}^{1.5}} \cdot \frac{a \cdot c}{{b}^{1.5}}}}\right)}^{-1} \]

   9. frac-times 94.7%

      \[\leadsto {\left(-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + \color{blue}{\frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{{b}^{1.5} \cdot {b}^{1.5}}}}\right)}^{-1} \]

   10. pow2 94.7%

       \[\leadsto {\left(-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{1.5} \cdot {b}^{1.5}}}\right)}^{-1} \]

   11. pow-prod-up 94.7%

       \[\leadsto {\left(-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + \frac{{\left(a \cdot c\right)}^{2}}{\color{blue}{{b}^{\left(1.5 + 1.5\right)}}}}\right)}^{-1} \]

   12. metadata-eval 94.7%

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

9. Applied egg-rr 94.7%

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

10. Taylor expanded in a around 0 94.9%

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

    1. +-commutative 94.9%

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

    2. mul-1-neg 94.9%

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

    3. unsub-neg 94.9%

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

12. Simplified 94.9%

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

13. Final simplification 94.9%

    \[\leadsto {\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1} \]
14. Add Preprocessing

Alternative 5: 90.0% 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 18.3%

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

   1. *-commutative 18.3%

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

3. Simplified 18.3%

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

5. Taylor expanded in b around inf 90.1%

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

   1. mul-1-neg 90.1%

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

   2. distribute-neg-frac 90.1%

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

7. Simplified 90.1%

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

8. Final simplification 90.1%

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

Reproduce

herbie shell --seed 2024010 
(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)))