Quadratic roots, narrow range

Percentage Accurate: 55.4% → 91.7%

Time: 13.6s

Alternatives: 7

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: 91.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(a \cdot c\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -200:\\ \;\;\;\;\frac{\sqrt{\frac{{b}^{6} - 64 \cdot \left({a}^{3} \cdot {c}^{3}\right)}{{b}^{4} + t_0 \cdot \mathsf{fma}\left(b, b, t_0\right)}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4} \cdot 20}{a \cdot {b}^{7}}, \frac{-a}{\frac{{b}^{3}}{c \cdot c}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 4.0 (* a c))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -200.0)
     (/
      (-
       (sqrt
        (/
         (- (pow b 6.0) (* 64.0 (* (pow a 3.0) (pow c 3.0))))
         (+ (pow b 4.0) (* t_0 (fma b b t_0)))))
       b)
      (* a 2.0))
     (fma
      -2.0
      (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
      (-
       (fma
        -0.25
        (/ (* (pow (* a c) 4.0) 20.0) (* a (pow b 7.0)))
        (/ (- a) (/ (pow b 3.0) (* c c))))
       (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = 4.0 * (a * c);
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -200.0) {
		tmp = (sqrt(((pow(b, 6.0) - (64.0 * (pow(a, 3.0) * pow(c, 3.0)))) / (pow(b, 4.0) + (t_0 * fma(b, b, t_0))))) - b) / (a * 2.0);
	} else {
		tmp = fma(-2.0, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), (fma(-0.25, ((pow((a * c), 4.0) * 20.0) / (a * pow(b, 7.0))), (-a / (pow(b, 3.0) / (c * c)))) - (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(4.0 * Float64(a * c))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -200.0)
		tmp = Float64(Float64(sqrt(Float64(Float64((b ^ 6.0) - Float64(64.0 * Float64((a ^ 3.0) * (c ^ 3.0)))) / Float64((b ^ 4.0) + Float64(t_0 * fma(b, b, t_0))))) - b) / Float64(a * 2.0));
	else
		tmp = fma(-2.0, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), Float64(fma(-0.25, Float64(Float64((Float64(a * c) ^ 4.0) * 20.0) / Float64(a * (b ^ 7.0))), Float64(Float64(-a) / Float64((b ^ 3.0) / Float64(c * c)))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -200.0], N[(N[(N[Sqrt[N[(N[(N[Power[b, 6.0], $MachinePrecision] - N[(64.0 * N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 4.0], $MachinePrecision] + N[(t$95$0 * N[(b * b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-a) / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $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)) < -200

   1. Initial program 92.5%

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

      1. flip3-- 91.5%

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

      2. pow2 91.5%

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

      3. pow-pow 91.9%

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

      4. metadata-eval 91.9%

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

      5. associate-*l* 91.9%

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

      6. pow2 91.9%

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

      7. pow2 91.9%

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

      8. pow-prod-up 92.6%

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

      9. metadata-eval 92.6%

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

      10. distribute-rgt-out 92.6%

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

      11. associate-*l* 92.6%

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

      12. +-commutative 92.6%

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

      13. fma-def 92.6%

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

      14. associate-*l* 92.6%

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

   3. Applied egg-rr 92.6%

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

   4. Taylor expanded in a around 0 92.8%

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

   if -200 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 53.1%

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

   2. Taylor expanded in b around inf 92.6%

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

   4. Simplified 92.6%

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

   5. Taylor expanded in a around 0 92.6%

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

      1. distribute-rgt-out 92.6%

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

      2. associate-*r* 92.6%

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

      3. metadata-eval 92.6%

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

      4. pow-sqr 92.6%

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

      5. unpow2 92.6%

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

      6. unpow2 92.6%

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

      7. metadata-eval 92.6%

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

      8. pow-sqr 92.6%

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

      9. unpow2 92.6%

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

      10. unpow2 92.6%

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

      11. swap-sqr 92.6%

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

      12. *-commutative 92.6%

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

      13. swap-sqr 92.6%

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

      14. *-commutative 92.6%

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

      15. swap-sqr 92.6%

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

      16. unpow2 92.6%

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

      17. unpow2 92.6%

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

      18. pow-sqr 92.6%

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

      19. *-commutative 92.6%

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

      20. metadata-eval 92.6%

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

      21. metadata-eval 92.6%

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

   7. Simplified 92.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -200:\\ \;\;\;\;\frac{\sqrt{\frac{{b}^{6} - 64 \cdot \left({a}^{3} \cdot {c}^{3}\right)}{{b}^{4} + \left(4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.25, \frac{{\left(a \cdot c\right)}^{4} \cdot 20}{a \cdot {b}^{7}}, \frac{-a}{\frac{{b}^{3}}{c \cdot c}}\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 2: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 24.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 24.2)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (fma -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a))) (/ (- c) b))
    (/ a (/ (pow b 3.0) (* c c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 24.2) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(-2.0, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), (-c / b)) - (a / (pow(b, 3.0) / (c * c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 24.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(-2.0, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), Float64(Float64(-c) / b)) - Float64(a / Float64((b ^ 3.0) / Float64(c * c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 24.2], 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[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 24.1999999999999993

   1. Initial program 81.1%

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

   3. Simplified 81.2%

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

   if 24.1999999999999993 < b

   1. Initial program 46.3%

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

   2. Taylor expanded in b around inf 93.6%

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

      1. associate-+r+ 93.6%

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

      2. mul-1-neg 93.6%

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

      3. unsub-neg 93.6%

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

      4. fma-def 93.6%

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

      5. *-commutative 93.6%

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

      6. associate-/l* 93.6%

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

      7. unpow2 93.6%

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

      8. mul-1-neg 93.6%

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

      9. distribute-neg-frac 93.6%

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

      10. associate-/l* 93.6%

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

      11. unpow2 93.6%

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

   4. Simplified 93.6%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 24.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \frac{-c}{b}\right) - \frac{a}{\frac{{b}^{3}}{c \cdot c}}\\ \end{array} \]

Alternative 3: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 24.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 24.2)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 24.2) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 24.2)
		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(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 24.2], 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[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 24.1999999999999993

   1. Initial program 81.1%

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

   3. Simplified 81.2%

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

   if 24.1999999999999993 < b

   1. Initial program 46.3%

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

   2. Taylor expanded in b around inf 89.3%

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

      1. distribute-lft-out 89.3%

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

      2. associate-/l* 89.3%

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

      3. pow-prod-down 89.3%

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

      4. *-commutative 89.3%

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

   4. Applied egg-rr 89.3%

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

      1. associate-/r/ 89.3%

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

      2. fma-def 89.4%

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

      3. *-commutative 89.4%

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

   6. Simplified 89.4%

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

   7. Taylor expanded in a around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. unsub-neg 89.5%

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

      3. associate-*r/ 89.5%

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

      4. mul-1-neg 89.5%

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

      5. cube-mult 89.5%

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

      6. unpow2 89.5%

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

      7. times-frac 89.5%

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

      8. unpow2 89.5%

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

      9. unpow2 89.5%

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

      10. times-frac 89.5%

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

      11. unpow2 89.5%

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

   9. Simplified 89.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 24.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 4: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 24.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 24.2)
   (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 24.2) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 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 <= 24.2d0) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a / b) * ((c / b) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 24.2) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * Math.pow((c / b), 2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 24.2:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a / b) * math.pow((c / b), 2.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 24.2)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 24.2)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((a / b) * ((c / b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 24.2], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 24.1999999999999993

   1. Initial program 81.1%

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

   3. Simplified 81.2%

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

      1. *-commutative 81.2%

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

      2. metadata-eval 81.2%

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

      3. distribute-lft-neg-in 81.2%

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

      4. distribute-rgt-neg-in 81.2%

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

      5. *-commutative 81.2%

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

      6. fma-neg 81.1%

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

      7. associate-*l* 81.1%

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

   5. Applied egg-rr 81.1%

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

   if 24.1999999999999993 < b

   1. Initial program 46.3%

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

   2. Taylor expanded in b around inf 89.3%

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

      1. distribute-lft-out 89.3%

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

      2. associate-/l* 89.3%

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

      3. pow-prod-down 89.3%

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

      4. *-commutative 89.3%

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

   4. Applied egg-rr 89.3%

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

      1. associate-/r/ 89.3%

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

      2. fma-def 89.4%

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

      3. *-commutative 89.4%

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

   6. Simplified 89.4%

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

   7. Taylor expanded in a around 0 89.5%

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

      1. mul-1-neg 89.5%

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

      2. unsub-neg 89.5%

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

      3. associate-*r/ 89.5%

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

      4. mul-1-neg 89.5%

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

      5. cube-mult 89.5%

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

      6. unpow2 89.5%

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

      7. times-frac 89.5%

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

      8. unpow2 89.5%

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

      9. unpow2 89.5%

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

      10. times-frac 89.5%

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

      11. unpow2 89.5%

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

   9. Simplified 89.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 24.2:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 5: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $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. Taylor expanded in b around inf 82.2%

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

   1. distribute-lft-out 82.2%

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

   2. associate-/l* 82.2%

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

   3. pow-prod-down 82.2%

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

   4. *-commutative 82.2%

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

4. Applied egg-rr 82.2%

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

   1. associate-/r/ 82.1%

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

   2. fma-def 82.2%

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

   3. *-commutative 82.2%

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

6. Simplified 82.2%

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

7. Taylor expanded in a around 0 82.3%

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

   1. mul-1-neg 82.3%

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

   2. unsub-neg 82.3%

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

   3. associate-*r/ 82.3%

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

   4. mul-1-neg 82.3%

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

   5. cube-mult 82.3%

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

   6. unpow2 82.3%

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

   7. times-frac 82.3%

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

   8. unpow2 82.3%

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

   9. unpow2 82.3%

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

   10. times-frac 82.3%

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

   11. unpow2 82.3%

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

9. Simplified 82.3%

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

10. Final simplification 82.3%

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

Alternative 6: 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 55.1%

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

2. Taylor expanded in b around inf 64.8%

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

   1. mul-1-neg 64.8%

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

   2. distribute-neg-frac 64.8%

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

4. Simplified 64.8%

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

5. Final simplification 64.8%

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

Alternative 7: 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 55.1%

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

2. Taylor expanded in b around -inf 11.6%

   \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
3. 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%

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

4. Simplified 11.6%

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

5. Taylor expanded in c around inf 1.6%

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

6. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2023279 
(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)))