Quadratic roots, narrow range

Percentage Accurate: 55.4% → 91.4%

Time: 13.4s

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: 55.4% 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: 91.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1)
   (/ (- (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)))
    (-
     (-
      (* -0.25 (* (pow (* c a) 4.0) (/ 20.0 (* 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) {
	double tmp;
	if (b <= 2.1) {
		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))) + (((-0.25 * (pow((c * a), 4.0) * (20.0 / (a * pow(b, 7.0))))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64(a * (b ^ 7.0))))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1], 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[(-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[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(a * N[Power[b, 7.0], $MachinePrecision]), $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. Split input into 2 regimes

2. if b < 2.10000000000000009

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

   3. Simplified 85.5%

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

   if 2.10000000000000009 < b

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in b around inf 94.4%

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

      1. unpow2 94.4%

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

      2. *-commutative 94.4%

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

      3. *-commutative 94.4%

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

      4. swap-sqr 94.4%

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

      5. pow-prod-down 94.4%

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

      6. pow-prod-down 94.4%

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

      7. pow-sqr 94.4%

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

      8. metadata-eval 94.4%

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

      9. metadata-eval 94.4%

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

   6. Applied egg-rr 94.4%

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

      1. pow-prod-down 94.4%

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

      2. metadata-eval 94.4%

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

      3. pow-sqr 94.4%

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

      4. pow-prod-down 94.4%

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

      5. pow-prod-down 94.4%

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

      6. pow1 94.4%

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

      7. metadata-eval 94.4%

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

      8. pow1 94.4%

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

      9. metadata-eval 94.4%

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

      10. sqr-pow 94.4%

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

      11. pow-prod-down 94.4%

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

   8. Applied egg-rr 94.4%

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

      1. unpow2 94.4%

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

      2. pow-sqr 94.4%

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

      3. metadata-eval 94.4%

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

   10. Simplified 94.4%

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

   11. Taylor expanded in c around 0 94.4%

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

       1. distribute-rgt-out 94.4%

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

       2. metadata-eval 94.4%

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

       3. associate-*r* 94.4%

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

       4. *-commutative 94.4%

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

   13. Simplified 94.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 2: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{{a}^{2} \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.1)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (- (/ -2.0 (/ (pow b 5.0) (* (pow a 2.0) (pow c 3.0)))) (/ c b))
    (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.1) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((-2.0 / (pow(b, 5.0) / (pow(a, 2.0) * pow(c, 3.0)))) - (c / b)) - ((a * pow(c, 2.0)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64((a ^ 2.0) * (c ^ 3.0)))) - Float64(c / b)) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.1], 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[(-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[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000009

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

   3. Simplified 85.5%

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

   if 2.10000000000000009 < b

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in b around inf 92.1%

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

      1. associate-+r+ 92.1%

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

      2. mul-1-neg 92.1%

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

      3. unsub-neg 92.1%

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

      4. mul-1-neg 92.1%

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

      5. unsub-neg 92.1%

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

      6. associate-*r/ 92.1%

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

      7. associate-/l* 92.1%

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

      8. *-commutative 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{{a}^{2} \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 3: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -1.8e-5) 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 <= -1.8e-5) {
		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 <= (-1.8d-5)) 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 <= -1.8e-5) {
		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 <= -1.8e-5:
		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 <= -1.8e-5)
		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 <= -1.8e-5)
		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, -1.8e-5], 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)) < -1.80000000000000005e-5

   1. Initial program 71.9%

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

   if -1.80000000000000005e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 37.9%

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

      1. *-commutative 37.9%

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

   3. Simplified 37.9%

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

   4. Taylor expanded in b around inf 79.8%

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

      1. mul-1-neg 79.8%

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

      2. distribute-neg-frac 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 4: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.7)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * pow(c, 2.0)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.7)
		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(Float64(a * (c ^ 2.0)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.7], 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[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7000000000000002

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

   3. Simplified 85.5%

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

   if 2.7000000000000002 < b

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in b around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. unsub-neg 87.1%

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

      3. mul-1-neg 87.1%

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

      4. distribute-neg-frac 87.1%

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

   6. Simplified 87.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 5: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.2)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.2) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * pow(c, 2.0)) / pow(b, 3.0));
	}
	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 <= 2.2d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.2) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.2:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.2)
		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(Float64(a * (c ^ 2.0)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.2)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((a * (c ^ 2.0)) / (b ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.2], 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[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2000000000000002

   1. Initial program 85.1%

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

   if 2.2000000000000002 < b

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in b around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. unsub-neg 87.1%

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

      3. mul-1-neg 87.1%

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

      4. distribute-neg-frac 87.1%

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

   6. Simplified 87.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 6: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c \cdot a}{b} + -2 \cdot \left(\frac{1}{b} \cdot {\left(\frac{c}{\frac{b}{a}}\right)}^{2}\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/
    (+ (* -2.0 (/ (* c a) b)) (* -2.0 (* (/ 1.0 b) (pow (/ c (/ b a)) 2.0))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((-2.0 * ((c * a) / b)) + (-2.0 * ((1.0 / b) * pow((c / (b / a)), 2.0)))) / (a * 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 <= 2.4d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (((-2.0d0) * ((c * a) / b)) + ((-2.0d0) * ((1.0d0 / b) * ((c / (b / a)) ** 2.0d0)))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((-2.0 * ((c * a) / b)) + (-2.0 * ((1.0 / b) * Math.pow((c / (b / a)), 2.0)))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = ((-2.0 * ((c * a) / b)) + (-2.0 * ((1.0 / b) * math.pow((c / (b / a)), 2.0)))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 * Float64(Float64(c * a) / b)) + Float64(-2.0 * Float64(Float64(1.0 / b) * (Float64(c / Float64(b / a)) ^ 2.0)))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.4)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = ((-2.0 * ((c * a) / b)) + (-2.0 * ((1.0 / b) * ((c / (b / a)) ^ 2.0)))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4], 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[(N[(-2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(1.0 / b), $MachinePrecision] * N[Power[N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.39999999999999991

   1. Initial program 85.1%

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

   if 2.39999999999999991 < b

   1. Initial program 50.3%

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

      1. *-commutative 50.3%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in b around inf 86.9%

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

      1. *-un-lft-identity 86.9%

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

      2. metadata-eval 86.9%

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

      3. cube-mult 86.9%

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

      4. times-frac 86.9%

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

      5. metadata-eval 86.9%

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

      6. unpow2 86.9%

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

      7. unpow2 86.9%

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

      8. swap-sqr 86.9%

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

      9. frac-times 86.9%

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

      10. associate-*l/ 86.9%

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

      11. associate-*l/ 86.9%

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

      12. pow1 86.9%

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

      13. associate-*l/ 86.9%

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

      14. metadata-eval 86.9%

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

      15. pow1 86.9%

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

      16. associate-*l/ 86.9%

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

      17. metadata-eval 86.9%

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

      18. pow-sqr 86.9%

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

      19. *-commutative 86.9%

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

      20. associate-/l* 86.9%

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

      21. metadata-eval 86.9%

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

      22. metadata-eval 86.9%

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

   6. Applied egg-rr 86.9%

      \[\leadsto \frac{-2 \cdot \frac{a \cdot c}{b} + -2 \cdot \color{blue}{\left(\frac{1}{b} \cdot {\left(\frac{c}{\frac{b}{a}}\right)}^{2}\right)}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c \cdot a}{b} + -2 \cdot \left(\frac{1}{b} \cdot {\left(\frac{c}{\frac{b}{a}}\right)}^{2}\right)}{a \cdot 2}\\ \end{array} \]

Alternative 7: 64.4% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

4. Taylor expanded in b around inf 64.7%

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

   1. mul-1-neg 64.7%

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

   2. distribute-neg-frac 64.7%

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

6. Simplified 64.7%

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

7. Final simplification 64.7%

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

Alternative 8: 1.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(c / 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 55.8%

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

   1. *-commutative 55.8%

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

3. Simplified 55.8%

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

4. Taylor expanded in b around -inf 11.6%

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

   1. +-commutative 11.6%

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

   2. mul-1-neg 11.6%

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

   3. unsub-neg 11.6%

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

6. Simplified 11.6%

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

7. Taylor expanded in c around inf 1.6%

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

8. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2023305 
(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)))