Quadratic roots, narrow range

Percentage Accurate: 55.8% → 91.6%

Time: 9.3s

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: 55.8% 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: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4}}{a \cdot c} \cdot 0 + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right)\right) - \frac{c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.22)
   (*
    (/ 0.5 a)
    (-
     (sqrt (+ (* (/ (pow b 4.0) (* a c)) 0.0) (fma b b (* c (* a -4.0)))))
     b))
   (-
    (-
     (fma
      -0.25
      (/ (pow a 3.0) (/ (pow b 7.0) (* (pow c 4.0) 20.0)))
      (* -2.0 (* (/ (pow c 3.0) (pow b 5.0)) (* a a))))
     (/ c b))
    (* a (/ c (/ (pow b 3.0) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.22) {
		tmp = (0.5 / a) * (sqrt((((pow(b, 4.0) / (a * c)) * 0.0) + fma(b, b, (c * (a * -4.0))))) - b);
	} else {
		tmp = (fma(-0.25, (pow(a, 3.0) / (pow(b, 7.0) / (pow(c, 4.0) * 20.0))), (-2.0 * ((pow(c, 3.0) / pow(b, 5.0)) * (a * a)))) - (c / b)) - (a * (c / (pow(b, 3.0) / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.22)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(Float64((b ^ 4.0) / Float64(a * c)) * 0.0) + fma(b, b, Float64(c * Float64(a * -4.0))))) - b));
	else
		tmp = Float64(Float64(fma(-0.25, Float64((a ^ 3.0) / Float64((b ^ 7.0) / Float64((c ^ 4.0) * 20.0))), Float64(-2.0 * Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * a)))) - Float64(c / b)) - Float64(a * Float64(c / Float64((b ^ 3.0) / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.22], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(N[Power[b, 4.0], $MachinePrecision] / N[(a * c), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision] + N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.21999999999999997

   1. Initial program 87.6%

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

      1. flip3-- 87.2%

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

      2. pow2 87.2%

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

      3. pow-pow 86.5%

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

      4. metadata-eval 86.5%

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

      5. associate-*l* 86.5%

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

      6. pow2 86.5%

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

      7. pow2 86.5%

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

      8. pow-prod-up 87.0%

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

      9. metadata-eval 87.0%

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

      10. distribute-rgt-out 87.0%

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

      11. associate-*l* 87.0%

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

      12. +-commutative 87.0%

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

      13. fma-def 87.0%

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

      14. associate-*l* 87.0%

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

   3. Applied egg-rr 87.0%

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

   4. Taylor expanded in a around -inf 72.8%

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

      1. div-inv 72.8%

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

      2. fma-def 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, 0.25 \cdot \frac{{b}^{4}}{c \cdot a} + \left({b}^{2} + -4 \cdot \left(c \cdot a\right)\right)\right)}}\right) \cdot \frac{1}{2 \cdot a} \]

      3. fma-def 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \color{blue}{\mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, {b}^{2} + -4 \cdot \left(c \cdot a\right)\right)}\right)}\right) \cdot \frac{1}{2 \cdot a} \]

      4. unpow2 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, \color{blue}{b \cdot b} + -4 \cdot \left(c \cdot a\right)\right)\right)}\right) \cdot \frac{1}{2 \cdot a} \]

      5. associate-*r* 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)\right)}\right) \cdot \frac{1}{2 \cdot a} \]

      6. *-commutative 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \cdot \frac{1}{\color{blue}{a \cdot 2}} \]

   6. Applied egg-rr 72.8%

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

      1. *-commutative 72.8%

         \[\leadsto \color{blue}{\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right)} \]

      2. *-commutative 72.8%

         \[\leadsto \frac{1}{\color{blue}{2 \cdot a}} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \]

      3. associate-/r* 72.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \]

      4. metadata-eval 72.8%

         \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \]

      5. +-commutative 72.8%

         \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)} + \left(-b\right)\right)} \]

      6. unsub-neg 72.8%

         \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)} - b\right)} \]

   8. Simplified 87.9%

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

   if 1.21999999999999997 < b

   1. Initial program 50.2%

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

   2. Taylor expanded in a around 0 94.0%

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

   4. Simplified 94.0%

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

   5. Taylor expanded in b around 0 94.0%

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

      1. associate-/l* 94.0%

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

      2. distribute-rgt-out 94.0%

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

      3. metadata-eval 94.0%

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

   7. Simplified 94.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4}}{a \cdot c} \cdot 0 + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4} \cdot 20}}, -2 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right)\right) - \frac{c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 2: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4}}{a \cdot c} \cdot 0 + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{-c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.3)
   (*
    (/ 0.5 a)
    (-
     (sqrt (+ (* (/ (pow b 4.0) (* a c)) 0.0) (fma b b (* c (* a -4.0)))))
     b))
   (-
    (fma -2.0 (* (/ (pow c 3.0) (pow b 5.0)) (* a a)) (/ (- c) b))
    (* a (/ c (/ (pow b 3.0) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.3) {
		tmp = (0.5 / a) * (sqrt((((pow(b, 4.0) / (a * c)) * 0.0) + fma(b, b, (c * (a * -4.0))))) - b);
	} else {
		tmp = fma(-2.0, ((pow(c, 3.0) / pow(b, 5.0)) * (a * a)), (-c / b)) - (a * (c / (pow(b, 3.0) / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.3)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(Float64((b ^ 4.0) / Float64(a * c)) * 0.0) + fma(b, b, Float64(c * Float64(a * -4.0))))) - b));
	else
		tmp = Float64(fma(-2.0, Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * a)), Float64(Float64(-c) / b)) - Float64(a * Float64(c / Float64((b ^ 3.0) / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.3], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(N[Power[b, 4.0], $MachinePrecision] / N[(a * c), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision] + N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.30000000000000004

   1. Initial program 87.6%

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

      1. flip3-- 87.2%

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

      2. pow2 87.2%

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

      3. pow-pow 86.5%

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

      4. metadata-eval 86.5%

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

      5. associate-*l* 86.5%

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

      6. pow2 86.5%

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

      7. pow2 86.5%

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

      8. pow-prod-up 87.0%

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

      9. metadata-eval 87.0%

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

      10. distribute-rgt-out 87.0%

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

      11. associate-*l* 87.0%

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

      12. +-commutative 87.0%

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

      13. fma-def 87.0%

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

      14. associate-*l* 87.0%

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

   3. Applied egg-rr 87.0%

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

   4. Taylor expanded in a around -inf 72.8%

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

      1. div-inv 72.8%

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

      2. fma-def 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, 0.25 \cdot \frac{{b}^{4}}{c \cdot a} + \left({b}^{2} + -4 \cdot \left(c \cdot a\right)\right)\right)}}\right) \cdot \frac{1}{2 \cdot a} \]

      3. fma-def 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \color{blue}{\mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, {b}^{2} + -4 \cdot \left(c \cdot a\right)\right)}\right)}\right) \cdot \frac{1}{2 \cdot a} \]

      4. unpow2 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, \color{blue}{b \cdot b} + -4 \cdot \left(c \cdot a\right)\right)\right)}\right) \cdot \frac{1}{2 \cdot a} \]

      5. associate-*r* 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)\right)}\right) \cdot \frac{1}{2 \cdot a} \]

      6. *-commutative 72.8%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \cdot \frac{1}{\color{blue}{a \cdot 2}} \]

   6. Applied egg-rr 72.8%

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

      1. *-commutative 72.8%

         \[\leadsto \color{blue}{\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right)} \]

      2. *-commutative 72.8%

         \[\leadsto \frac{1}{\color{blue}{2 \cdot a}} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \]

      3. associate-/r* 72.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \]

      4. metadata-eval 72.8%

         \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \]

      5. +-commutative 72.8%

         \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)} + \left(-b\right)\right)} \]

      6. unsub-neg 72.8%

         \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)} - b\right)} \]

   8. Simplified 87.9%

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

   if 1.30000000000000004 < b

   1. Initial program 50.2%

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

   2. Taylor expanded in b around inf 91.6%

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

      1. +-commutative 91.6%

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

      2. mul-1-neg 91.6%

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

      3. unsub-neg 91.6%

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

      4. +-commutative 91.6%

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

      5. fma-def 91.6%

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

      6. associate-/l* 91.6%

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

      7. associate-/r/ 91.6%

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

      8. unpow2 91.6%

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

      9. mul-1-neg 91.6%

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

      10. distribute-neg-frac 91.6%

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

      11. associate-/l* 91.6%

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

      12. associate-/r/ 91.6%

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

      13. unpow2 91.6%

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

      14. associate-/l* 91.6%

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

   4. Simplified 91.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4}}{a \cdot c} \cdot 0 + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \frac{-c}{b}\right) - a \cdot \frac{c}{\frac{{b}^{3}}{c}}\\ \end{array} \]

Alternative 3: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4}}{a \cdot c} \cdot 0 + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{{b}^{3}}{c}} \cdot \left(-a\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.4)
   (*
    (/ 0.5 a)
    (-
     (sqrt (+ (* (/ (pow b 4.0) (* a c)) 0.0) (fma b b (* c (* a -4.0)))))
     b))
   (- (* (/ c (/ (pow b 3.0) c)) (- a)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.4) {
		tmp = (0.5 / a) * (sqrt((((pow(b, 4.0) / (a * c)) * 0.0) + fma(b, b, (c * (a * -4.0))))) - b);
	} else {
		tmp = ((c / (pow(b, 3.0) / c)) * -a) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.4)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(Float64((b ^ 4.0) / Float64(a * c)) * 0.0) + fma(b, b, Float64(c * Float64(a * -4.0))))) - b));
	else
		tmp = Float64(Float64(Float64(c / Float64((b ^ 3.0) / c)) * Float64(-a)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.4], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(N[Power[b, 4.0], $MachinePrecision] / N[(a * c), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision] + N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   1. Initial program 84.2%

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

      1. flip3-- 83.8%

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

      2. pow2 83.8%

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

      3. pow-pow 83.3%

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

      4. metadata-eval 83.3%

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

      5. associate-*l* 83.3%

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

      6. pow2 83.3%

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

      7. pow2 83.3%

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

      8. pow-prod-up 83.8%

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

      9. metadata-eval 83.8%

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

      10. distribute-rgt-out 83.8%

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

      11. associate-*l* 83.8%

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

      12. +-commutative 83.8%

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

      13. fma-def 83.8%

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

      14. associate-*l* 83.8%

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

   3. Applied egg-rr 83.8%

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

   4. Taylor expanded in a around -inf 67.5%

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

      1. div-inv 67.5%

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

      2. fma-def 67.5%

         \[\leadsto \left(\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, 0.25 \cdot \frac{{b}^{4}}{c \cdot a} + \left({b}^{2} + -4 \cdot \left(c \cdot a\right)\right)\right)}}\right) \cdot \frac{1}{2 \cdot a} \]

      3. fma-def 67.5%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \color{blue}{\mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, {b}^{2} + -4 \cdot \left(c \cdot a\right)\right)}\right)}\right) \cdot \frac{1}{2 \cdot a} \]

      4. unpow2 67.5%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, \color{blue}{b \cdot b} + -4 \cdot \left(c \cdot a\right)\right)\right)}\right) \cdot \frac{1}{2 \cdot a} \]

      5. associate-*r* 67.5%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)\right)}\right) \cdot \frac{1}{2 \cdot a} \]

      6. *-commutative 67.5%

         \[\leadsto \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \cdot \frac{1}{\color{blue}{a \cdot 2}} \]

   6. Applied egg-rr 67.5%

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

      1. *-commutative 67.5%

         \[\leadsto \color{blue}{\frac{1}{a \cdot 2} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right)} \]

      2. *-commutative 67.5%

         \[\leadsto \frac{1}{\color{blue}{2 \cdot a}} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \]

      3. associate-/r* 67.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \]

      4. metadata-eval 67.5%

         \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)}\right) \]

      5. +-commutative 67.5%

         \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)} + \left(-b\right)\right)} \]

      6. unsub-neg 67.5%

         \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(-0.25, \frac{{b}^{4}}{c \cdot a}, \mathsf{fma}\left(0.25, \frac{{b}^{4}}{c \cdot a}, b \cdot b + \left(-4 \cdot c\right) \cdot a\right)\right)} - b\right)} \]

   8. Simplified 84.4%

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

   if 7.4000000000000004 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. mul-1-neg 87.5%

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

      5. distribute-neg-frac 87.5%

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

      6. associate-/l* 87.5%

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

      7. associate-/r/ 87.5%

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

      8. unpow2 87.5%

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

      9. associate-/l* 87.5%

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

   4. Simplified 87.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\frac{{b}^{4}}{a \cdot c} \cdot 0 + \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{{b}^{3}}{c}} \cdot \left(-a\right) - \frac{c}{b}\\ \end{array} \]

Alternative 4: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{{b}^{3}}{c}} \cdot \left(-a\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.5)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (* (/ c (/ (pow b 3.0) c)) (- a)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.5) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((c / (pow(b, 3.0) / c)) * -a) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(c / Float64((b ^ 3.0) / c)) * Float64(-a)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.5], 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[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5

   1. Initial program 84.2%

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

   3. Simplified 84.4%

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

   if 7.5 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. mul-1-neg 87.5%

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

      5. distribute-neg-frac 87.5%

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

      6. associate-/l* 87.5%

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

      7. associate-/r/ 87.5%

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

      8. unpow2 87.5%

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

      9. associate-/l* 87.5%

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

   4. Simplified 87.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{{b}^{3}}{c}} \cdot \left(-a\right) - \frac{c}{b}\\ \end{array} \]

Alternative 5: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{{b}^{3}}{c}} \cdot \left(-a\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.4)
   (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
   (- (* (/ c (/ (pow b 3.0) c)) (- a)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.4) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = ((c / (pow(b, 3.0) / c)) * -a) - (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) :: tmp
    if (b <= 7.4d0) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = ((c / ((b ** 3.0d0) / c)) * -a) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.4) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = ((c / (Math.pow(b, 3.0) / c)) * -a) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.4:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = ((c / (math.pow(b, 3.0) / c)) * -a) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.4)
		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 / Float64((b ^ 3.0) / c)) * Float64(-a)) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.4)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = ((c / ((b ^ 3.0) / c)) * -a) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.4], 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[(N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.4000000000000004

   1. Initial program 84.2%

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

   3. Simplified 84.4%

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

      1. *-commutative 84.4%

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

      2. metadata-eval 84.4%

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

      3. distribute-lft-neg-in 84.4%

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

      4. distribute-rgt-neg-in 84.4%

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

      5. *-commutative 84.4%

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

      6. fma-neg 84.2%

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

      7. associate-*l* 84.2%

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

   5. Applied egg-rr 84.2%

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

   if 7.4000000000000004 < b

   1. Initial program 48.0%

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

   2. Taylor expanded in b around inf 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. mul-1-neg 87.5%

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

      5. distribute-neg-frac 87.5%

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

      6. associate-/l* 87.5%

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

      7. associate-/r/ 87.5%

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

      8. unpow2 87.5%

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

      9. associate-/l* 87.5%

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

   4. Simplified 87.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{{b}^{3}}{c}} \cdot \left(-a\right) - \frac{c}{b}\\ \end{array} \]

Alternative 6: 81.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in b around inf 79.4%

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

   1. +-commutative 79.4%

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

   2. mul-1-neg 79.4%

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

   3. unsub-neg 79.4%

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

   4. mul-1-neg 79.4%

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

   5. distribute-neg-frac 79.4%

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

   6. associate-/l* 79.4%

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

   7. associate-/r/ 79.4%

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

   8. unpow2 79.4%

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

   9. associate-/l* 79.4%

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

4. Simplified 79.4%

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

5. Final simplification 79.4%

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

Alternative 7: 64.0% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in b around inf 62.2%

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

   1. mul-1-neg 62.2%

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

   2. distribute-neg-frac 62.2%

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

4. Simplified 62.2%

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

5. Final simplification 62.2%

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

Reproduce

herbie shell --seed 2023264 
(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)))