Quadratic roots, narrow range

Percentage Accurate: 56.0% → 90.5%

Time: 13.0s

Alternatives: 8

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 56.0% 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: 90.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 4 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (*
     -0.25
     (/
      (+ (* 16.0 (* (pow a 4.0) (pow c 4.0))) (* 4.0 (pow (* a c) 4.0)))
      (* a (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 * (((16.0 * (pow(a, 4.0) * pow(c, 4.0))) + (4.0 * pow((a * c), 4.0))) / (a * 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) * (((16.0d0 * ((a ** 4.0d0) * (c ** 4.0d0))) + (4.0d0 * ((a * c) ** 4.0d0))) / (a * (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 * (((16.0 * (Math.pow(a, 4.0) * Math.pow(c, 4.0))) + (4.0 * Math.pow((a * c), 4.0))) / (a * 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 * (((16.0 * (math.pow(a, 4.0) * math.pow(c, 4.0))) + (4.0 * math.pow((a * c), 4.0))) / (a * 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(16.0 * Float64((a ^ 4.0) * (c ^ 4.0))) + Float64(4.0 * (Float64(a * c) ^ 4.0))) / Float64(a * (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 * (((16.0 * ((a ^ 4.0) * (c ^ 4.0))) + (4.0 * ((a * c) ^ 4.0))) / (a * (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[(16.0 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 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 55.1%

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

   1. *-commutative 55.1%

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

3. Simplified 55.1%

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

4. Taylor expanded in b around inf 91.5%

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

   1. *-commutative 91.5%

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

   2. unpow-prod-down 91.5%

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

   3. pow-prod-down 91.5%

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

   4. pow-pow 91.5%

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

   5. metadata-eval 91.5%

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

   6. metadata-eval 91.5%

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

6. Applied egg-rr 91.5%

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

7. Final simplification 91.5%

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

Alternative 2: 90.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot {\left(a \cdot c\right)}^{2}\\ \frac{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{t_0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot t_0\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (pow (* a c) 2.0))))
   (/
    (fma
     -2.0
     (/ c (/ b a))
     (*
      -0.5
      (+
       (+
        (/ t_0 (pow b 3.0))
        (/ (fma 2.0 (* a (* c t_0)) (* a (* c 0.0))) (pow b 5.0)))
       (/ (pow (* a c) 4.0) (/ (pow b 7.0) 20.0)))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * pow((a * c), 2.0);
	return fma(-2.0, (c / (b / a)), (-0.5 * (((t_0 / pow(b, 3.0)) + (fma(2.0, (a * (c * t_0)), (a * (c * 0.0))) / pow(b, 5.0))) + (pow((a * c), 4.0) / (pow(b, 7.0) / 20.0))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * (Float64(a * c) ^ 2.0))
	return Float64(fma(-2.0, Float64(c / Float64(b / a)), Float64(-0.5 * Float64(Float64(Float64(t_0 / (b ^ 3.0)) + Float64(fma(2.0, Float64(a * Float64(c * t_0)), Float64(a * Float64(c * 0.0))) / (b ^ 5.0))) + Float64((Float64(a * c) ^ 4.0) / Float64((b ^ 7.0) / 20.0))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(-2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(N[(t$95$0 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(a * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 55.1%

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

   1. *-commutative 55.1%

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

3. Simplified 55.1%

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

   1. flip3-- 54.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

   2. sqrt-div 54.3%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2} \]

   3. pow2 54.3%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{\color{blue}{\left({b}^{2}\right)}}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   4. pow-pow 54.1%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{\color{blue}{{b}^{\left(2 \cdot 3\right)}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   5. metadata-eval 54.1%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{\color{blue}{6}} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   6. associate-*l* 54.1%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - {\color{blue}{\left(4 \cdot \left(a \cdot c\right)\right)}}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   7. unpow-prod-down 54.1%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{{4}^{3} \cdot {\left(a \cdot c\right)}^{3}}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   8. metadata-eval 54.1%

      \[\leadsto \frac{\left(-b\right) + \frac{\sqrt{{b}^{6} - \color{blue}{64} \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

5. Applied egg-rr 54.4%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{{b}^{6} - 64 \cdot {\left(a \cdot c\right)}^{3}}}{\sqrt{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}}}}{a \cdot 2} \]

6. Taylor expanded in b around inf 91.3%

   \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + \left(-0.5 \cdot \frac{-16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(2 \cdot \left(a \cdot \left(c \cdot \left(-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(16 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{{b}^{7}} + \left(-0.5 \cdot \frac{-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{{b}^{3}} + -0.5 \cdot \frac{-4 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 2 \cdot \left(a \cdot \left(c \cdot \left(-16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-2 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{5}}\right)\right)}}{a \cdot 2} \]

8. Simplified 91.3%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{\left({\left(0 + -0.5 \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right)}^{2} + 0\right) + \mathsf{fma}\left(a \cdot 2, c \cdot \mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right), 0\right)}{{b}^{7}}\right)\right)}}{a \cdot 2} \]

9. Taylor expanded in a around 0 91.3%

   \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \color{blue}{\frac{{a}^{4} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right)}{a \cdot 2} \]
10. Step-by-step derivation

    1. *-commutative 91.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{\color{blue}{\left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right) \cdot {a}^{4}}}{{b}^{7}}\right)\right)}{a \cdot 2} \]

    2. distribute-rgt-out 91.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{\color{blue}{\left({c}^{4} \cdot \left(4 + 16\right)\right)} \cdot {a}^{4}}{{b}^{7}}\right)\right)}{a \cdot 2} \]

    3. associate-*l* 91.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{\color{blue}{{c}^{4} \cdot \left(\left(4 + 16\right) \cdot {a}^{4}\right)}}{{b}^{7}}\right)\right)}{a \cdot 2} \]

    4. *-commutative 91.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{{b}^{7}}\right)\right)}{a \cdot 2} \]

    5. distribute-rgt-out 91.3%

       \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{{c}^{4} \cdot \color{blue}{\left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}}{{b}^{7}}\right)\right)}{a \cdot 2} \]

11. Simplified 91.3%

    \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2} + 0}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2} + 0\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}}\right)\right)}{a \cdot 2} \]

12. Final simplification 91.3%

    \[\leadsto \frac{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, -0.5 \cdot \left(\left(\frac{4 \cdot {\left(a \cdot c\right)}^{2}}{{b}^{3}} + \frac{\mathsf{fma}\left(2, a \cdot \left(c \cdot \left(4 \cdot {\left(a \cdot c\right)}^{2}\right)\right), a \cdot \left(c \cdot 0\right)\right)}{{b}^{5}}\right) + \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \]

Alternative 3: 90.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (* -4.0 (/ (* (pow c 3.0) (pow a 3.0)) (pow b 5.0)))
   (+
    (* -2.0 (/ (* a c) b))
    (+
     (* -2.0 (/ (* (pow a 2.0) (pow c 2.0)) (pow b 3.0)))
     (* -0.5 (/ (pow (* a c) 4.0) (/ (pow b 7.0) 20.0))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return ((-4.0 * ((pow(c, 3.0) * pow(a, 3.0)) / pow(b, 5.0))) + ((-2.0 * ((a * c) / b)) + ((-2.0 * ((pow(a, 2.0) * pow(c, 2.0)) / pow(b, 3.0))) + (-0.5 * (pow((a * c), 4.0) / (pow(b, 7.0) / 20.0)))))) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-4.0d0) * (((c ** 3.0d0) * (a ** 3.0d0)) / (b ** 5.0d0))) + (((-2.0d0) * ((a * c) / b)) + (((-2.0d0) * (((a ** 2.0d0) * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.5d0) * (((a * c) ** 4.0d0) / ((b ** 7.0d0) / 20.0d0)))))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return ((-4.0 * ((Math.pow(c, 3.0) * Math.pow(a, 3.0)) / Math.pow(b, 5.0))) + ((-2.0 * ((a * c) / b)) + ((-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.5 * (Math.pow((a * c), 4.0) / (Math.pow(b, 7.0) / 20.0)))))) / (a * 2.0);
}

Python

def code(a, b, c):
	return ((-4.0 * ((math.pow(c, 3.0) * math.pow(a, 3.0)) / math.pow(b, 5.0))) + ((-2.0 * ((a * c) / b)) + ((-2.0 * ((math.pow(a, 2.0) * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.5 * (math.pow((a * c), 4.0) / (math.pow(b, 7.0) / 20.0)))))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-4.0 * Float64(Float64((c ^ 3.0) * (a ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-2.0 * Float64(Float64(a * c) / b)) + Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.5 * Float64((Float64(a * c) ^ 4.0) / Float64((b ^ 7.0) / 20.0)))))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-4.0 * (((c ^ 3.0) * (a ^ 3.0)) / (b ^ 5.0))) + ((-2.0 * ((a * c) / b)) + ((-2.0 * (((a ^ 2.0) * (c ^ 2.0)) / (b ^ 3.0))) + (-0.5 * (((a * c) ^ 4.0) / ((b ^ 7.0) / 20.0)))))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-4.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.1%

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

   1. *-commutative 55.1%

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

3. Simplified 55.1%

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

4. Taylor expanded in b around inf 91.3%

   \[\leadsto \frac{\color{blue}{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{{b}^{7}}\right)\right)}}{a \cdot 2} \]

5. Taylor expanded in a around 0 91.3%

   \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \color{blue}{\frac{{a}^{4} \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}}\right)\right)}{a \cdot 2} \]
6. Step-by-step derivation

7. Simplified 91.3%

   \[\leadsto \frac{-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}}\right)\right)}{a \cdot 2} \]

8. Final simplification 91.3%

   \[\leadsto \frac{-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(-2 \cdot \frac{a \cdot c}{b} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{{b}^{7}}{20}}\right)\right)}{a \cdot 2} \]

Alternative 4: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.155:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.155)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (- (/ (* -2.0 (* (pow a 2.0) (pow c 3.0))) (pow b 5.0)) (/ c b))
    (/ a (/ (pow b 3.0) (pow c 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.155) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (pow(a, 2.0) * pow(c, 3.0))) / pow(b, 5.0)) - (c / b)) - (a / (pow(b, 3.0) / pow(c, 2.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.155)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((a ^ 2.0) * (c ^ 3.0))) / (b ^ 5.0)) - Float64(c / b)) - Float64(a / Float64((b ^ 3.0) / (c ^ 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.155], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $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. Split input into 2 regimes

2. if b < 0.154999999999999999

   1. Initial program 84.1%

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

      1. sqr-neg 84.1%

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

      2. +-commutative 84.1%

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

      3. unsub-neg 84.1%

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

      4. sqr-neg 84.1%

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

      5. fma-neg 84.2%

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

      6. distribute-lft-neg-in 84.2%

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

      7. *-commutative 84.2%

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

      8. *-commutative 84.2%

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

      9. distribute-rgt-neg-in 84.2%

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

      10. metadata-eval 84.2%

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

      11. *-commutative 84.2%

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

   3. Simplified 84.2%

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

   if 0.154999999999999999 < b

   1. Initial program 51.2%

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

      1. *-commutative 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in b around inf 90.1%

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

      1. associate-+r+ 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

      4. mul-1-neg 90.1%

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

      5. unsub-neg 90.1%

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

      6. associate-*r/ 90.1%

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

      7. *-commutative 90.1%

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

      8. associate-/l* 90.1%

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

   6. Simplified 90.1%

      \[\leadsto \color{blue}{\left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.155:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \]

Alternative 5: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -4e-6) t_0 (/ (- c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -4e-6) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-4d-6)) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -4e-6) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -4e-6:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -4e-6)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -4e-6)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-6], t$95$0, N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -3.99999999999999982e-6

   1. Initial program 73.0%

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

   if -3.99999999999999982e-6 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 33.1%

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

      1. *-commutative 33.1%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in b around inf 82.6%

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

      1. mul-1-neg 82.6%

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

      2. distribute-neg-frac 82.6%

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

   6. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 6: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.2)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (/ a (/ (pow b 3.0) (pow c 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.2) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - (a / (pow(b, 3.0) / pow(c, 2.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(a / Float64((b ^ 3.0) / (c ^ 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.2], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 14.199999999999999

   1. Initial program 79.3%

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

      1. sqr-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. unsub-neg 79.3%

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

      4. sqr-neg 79.3%

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

      5. fma-neg 79.4%

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

      6. distribute-lft-neg-in 79.4%

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

      7. *-commutative 79.4%

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

      8. *-commutative 79.4%

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

      9. distribute-rgt-neg-in 79.4%

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

      10. metadata-eval 79.4%

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

      11. *-commutative 79.4%

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

   3. Simplified 79.4%

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

   if 14.199999999999999 < b

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in b around inf 86.7%

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

      1. mul-1-neg 86.7%

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

      2. unsub-neg 86.7%

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

      3. mul-1-neg 86.7%

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

      4. distribute-neg-frac 86.7%

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

      5. associate-/l* 86.7%

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

   6. Simplified 86.7%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \]

Alternative 7: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 14.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 14.5)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (/ a (/ (pow b 3.0) (pow c 2.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.5) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - (a / (pow(b, 3.0) / pow(c, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 14.5d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - (a / ((b ** 3.0d0) / (c ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 14.5) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - (a / (Math.pow(b, 3.0) / Math.pow(c, 2.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 14.5:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - (a / (math.pow(b, 3.0) / math.pow(c, 2.0)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 14.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(a / Float64((b ^ 3.0) / (c ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 14.5)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - (a / ((b ^ 3.0) / (c ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 14.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 14.5

   1. Initial program 79.3%

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

   if 14.5 < b

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

   3. Simplified 47.5%

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

   4. Taylor expanded in b around inf 86.7%

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

      1. mul-1-neg 86.7%

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

      2. unsub-neg 86.7%

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

      3. mul-1-neg 86.7%

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

      4. distribute-neg-frac 86.7%

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

      5. associate-/l* 86.7%

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

   6. Simplified 86.7%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 14.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}\\ \end{array} \]

Alternative 8: 63.9% 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 55.1%

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

   1. *-commutative 55.1%

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

3. Simplified 55.1%

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

4. Taylor expanded in b around inf 64.6%

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

   1. mul-1-neg 64.6%

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

   2. distribute-neg-frac 64.6%

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

6. Simplified 64.6%

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

7. Final simplification 64.6%

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

Reproduce

herbie shell --seed 2023339 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))