Quadratic roots, narrow range

Percentage Accurate: 55.3% → 92.3%

Time: 11.6s

Alternatives: 9

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: 55.3% 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: 92.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{c}^{4}}{{b}^{6}} \cdot 20\\ t_1 := {\left(\frac{c}{b}\right)}^{3}\\ t_2 := b \cdot b - \left(a \cdot c\right) \cdot 4\\ t_3 := \mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\\ t_4 := \frac{c \cdot c}{b}\\ t_5 := t_2 + \left(b \cdot b + b \cdot \sqrt{t_2}\right)\\ t_6 := \frac{c}{\frac{b}{0}}\\ \mathbf{if}\;b \leq 0.025:\\ \;\;\;\;\frac{\frac{\frac{{t_3}^{3} - {b}^{6}}{{b}^{3} + {t_3}^{1.5}}}{t_5}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, \left(b \cdot c\right) \cdot -6, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-2, t_4, \mathsf{fma}\left(8, t_4, b \cdot 0\right)\right), \mathsf{fma}\left({a}^{3}, \mathsf{fma}\left(-4, t_1, \mathsf{fma}\left(-2, t_6, \mathsf{fma}\left(8, t_1, b \cdot 0\right)\right)\right), {a}^{4} \cdot \mathsf{fma}\left(-2, t_6, \mathsf{fma}\left(-2, \frac{c \cdot c}{\frac{{b}^{3}}{0}}, \mathsf{fma}\left(-0.5, b \cdot t_0, \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{5}}, b \cdot \left(t_0 - t_0\right)\right)\right)\right)\right)\right)\right)\right)}{t_5}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (pow c 4.0) (pow b 6.0)) 20.0))
        (t_1 (pow (/ c b) 3.0))
        (t_2 (- (* b b) (* (* a c) 4.0)))
        (t_3 (fma b b (* -4.0 (* a c))))
        (t_4 (/ (* c c) b))
        (t_5 (+ t_2 (+ (* b b) (* b (sqrt t_2)))))
        (t_6 (/ c (/ b 0.0))))
   (if (<= b 0.025)
     (/
      (/ (/ (- (pow t_3 3.0) (pow b 6.0)) (+ (pow b 3.0) (pow t_3 1.5))) t_5)
      (* a 2.0))
     (/
      (/
       (fma
        a
        (* (* b c) -6.0)
        (fma
         (* a a)
         (fma -2.0 t_4 (fma 8.0 t_4 (* b 0.0)))
         (fma
          (pow a 3.0)
          (fma -4.0 t_1 (fma -2.0 t_6 (fma 8.0 t_1 (* b 0.0))))
          (*
           (pow a 4.0)
           (fma
            -2.0
            t_6
            (fma
             -2.0
             (/ (* c c) (/ (pow b 3.0) 0.0))
             (fma
              -0.5
              (* b t_0)
              (fma 16.0 (/ (pow c 4.0) (pow b 5.0)) (* b (- t_0 t_0))))))))))
       t_5)
      (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (pow(c, 4.0) / pow(b, 6.0)) * 20.0;
	double t_1 = pow((c / b), 3.0);
	double t_2 = (b * b) - ((a * c) * 4.0);
	double t_3 = fma(b, b, (-4.0 * (a * c)));
	double t_4 = (c * c) / b;
	double t_5 = t_2 + ((b * b) + (b * sqrt(t_2)));
	double t_6 = c / (b / 0.0);
	double tmp;
	if (b <= 0.025) {
		tmp = (((pow(t_3, 3.0) - pow(b, 6.0)) / (pow(b, 3.0) + pow(t_3, 1.5))) / t_5) / (a * 2.0);
	} else {
		tmp = (fma(a, ((b * c) * -6.0), fma((a * a), fma(-2.0, t_4, fma(8.0, t_4, (b * 0.0))), fma(pow(a, 3.0), fma(-4.0, t_1, fma(-2.0, t_6, fma(8.0, t_1, (b * 0.0)))), (pow(a, 4.0) * fma(-2.0, t_6, fma(-2.0, ((c * c) / (pow(b, 3.0) / 0.0)), fma(-0.5, (b * t_0), fma(16.0, (pow(c, 4.0) / pow(b, 5.0)), (b * (t_0 - t_0)))))))))) / t_5) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * 20.0)
	t_1 = Float64(c / b) ^ 3.0
	t_2 = Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))
	t_3 = fma(b, b, Float64(-4.0 * Float64(a * c)))
	t_4 = Float64(Float64(c * c) / b)
	t_5 = Float64(t_2 + Float64(Float64(b * b) + Float64(b * sqrt(t_2))))
	t_6 = Float64(c / Float64(b / 0.0))
	tmp = 0.0
	if (b <= 0.025)
		tmp = Float64(Float64(Float64(Float64((t_3 ^ 3.0) - (b ^ 6.0)) / Float64((b ^ 3.0) + (t_3 ^ 1.5))) / t_5) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(a, Float64(Float64(b * c) * -6.0), fma(Float64(a * a), fma(-2.0, t_4, fma(8.0, t_4, Float64(b * 0.0))), fma((a ^ 3.0), fma(-4.0, t_1, fma(-2.0, t_6, fma(8.0, t_1, Float64(b * 0.0)))), Float64((a ^ 4.0) * fma(-2.0, t_6, fma(-2.0, Float64(Float64(c * c) / Float64((b ^ 3.0) / 0.0)), fma(-0.5, Float64(b * t_0), fma(16.0, Float64((c ^ 4.0) / (b ^ 5.0)), Float64(b * Float64(t_0 - t_0)))))))))) / t_5) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(c / b), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 + N[(N[(b * b), $MachinePrecision] + N[(b * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(c / N[(b / 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.025], N[(N[(N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] - N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] + N[Power[t$95$3, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(N[(b * c), $MachinePrecision] * -6.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(-2.0 * t$95$4 + N[(8.0 * t$95$4 + N[(b * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 3.0], $MachinePrecision] * N[(-4.0 * t$95$1 + N[(-2.0 * t$95$6 + N[(8.0 * t$95$1 + N[(b * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 4.0], $MachinePrecision] * N[(-2.0 * t$95$6 + N[(-2.0 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(b * t$95$0), $MachinePrecision] + N[(16.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(b * N[(t$95$0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.025000000000000001

   1. Initial program 89.1%

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

   3. Simplified 89.1%

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

      1. *-commutative 89.1%

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

      2. metadata-eval 89.1%

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

      3. distribute-lft-neg-in 89.1%

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

      4. distribute-rgt-neg-in 89.1%

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

      5. *-commutative 89.1%

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

      6. fma-neg 89.1%

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

      7. associate-*l* 89.1%

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

   5. Applied egg-rr 89.1%

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

      1. flip3-- 88.9%

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

      2. add-sqr-sqrt 89.1%

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

   7. Applied egg-rr 89.1%

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

      1. flip-- 88.9%

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

   9. Applied egg-rr 89.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} \cdot {\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. pow-sqr 89.3%

          \[\leadsto \frac{\frac{\frac{\color{blue}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{\left(2 \cdot 1.5\right)}} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       2. fma-def 90.3%

          \[\leadsto \frac{\frac{\frac{{\color{blue}{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}}^{\left(2 \cdot 1.5\right)} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       3. *-commutative 90.3%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)\right)}^{\left(2 \cdot 1.5\right)} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       4. metadata-eval 90.3%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{\color{blue}{3}} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       5. pow-sqr 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - \color{blue}{{b}^{\left(2 \cdot 3\right)}}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       6. metadata-eval 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{\color{blue}{6}}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       7. +-commutative 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{\color{blue}{{b}^{3} + {\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       8. fma-def 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{{b}^{3} + {\color{blue}{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}}^{1.5}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       9. *-commutative 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{{b}^{3} + {\left(\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)\right)}^{1.5}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

   11. Simplified 90.0%

       \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{{b}^{3} + {\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{1.5}}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

   if 0.025000000000000001 < b

   1. Initial program 52.8%

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

   3. Simplified 52.8%

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

      1. *-commutative 52.8%

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

      2. metadata-eval 52.8%

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

      3. distribute-lft-neg-in 52.8%

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

      4. distribute-rgt-neg-in 52.8%

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

      5. *-commutative 52.8%

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

      6. fma-neg 52.8%

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

      7. associate-*l* 52.8%

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

   5. Applied egg-rr 52.8%

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

      1. flip3-- 52.6%

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

      2. add-sqr-sqrt 52.6%

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

   7. Applied egg-rr 52.6%

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

   8. Taylor expanded in a around 0 93.5%

      \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(-4 \cdot \left(b \cdot c\right) + -2 \cdot \left(b \cdot c\right)\right) + \left({a}^{2} \cdot \left(-2 \cdot \frac{{c}^{2}}{b} + \left(8 \cdot \frac{{c}^{2}}{b} + b \cdot \left(-4 \cdot \frac{{c}^{2}}{{b}^{2}} + 4 \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right) + \left({a}^{3} \cdot \left(-4 \cdot \frac{{c}^{3}}{{b}^{3}} + \left(-2 \cdot \frac{c \cdot \left(-4 \cdot \frac{{c}^{2}}{{b}^{2}} + 4 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{b} + \left(8 \cdot \frac{{c}^{3}}{{b}^{3}} + b \cdot \left(-8 \cdot \frac{{c}^{3}}{{b}^{4}} + 8 \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)\right)\right) + {a}^{4} \cdot \left(-2 \cdot \frac{c \cdot \left(-8 \cdot \frac{{c}^{3}}{{b}^{4}} + 8 \cdot \frac{{c}^{3}}{{b}^{4}}\right)}{b} + \left(-2 \cdot \frac{{c}^{2} \cdot \left(-4 \cdot \frac{{c}^{2}}{{b}^{2}} + 4 \cdot \frac{{c}^{2}}{{b}^{2}}\right)}{{b}^{3}} + \left(-0.5 \cdot \left(b \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)\right) + \left(16 \cdot \frac{{c}^{4}}{{b}^{5}} + b \cdot \left(-1 \cdot \left(16 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-2 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right) + \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

   10. Simplified 93.5%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(a, \left(b \cdot c\right) \cdot -6, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-2, \frac{c \cdot c}{b}, \mathsf{fma}\left(8, \frac{c \cdot c}{b}, b \cdot 0\right)\right), \mathsf{fma}\left({a}^{3}, \mathsf{fma}\left(-4, {\left(\frac{c}{b}\right)}^{3}, \mathsf{fma}\left(-2, \frac{c}{\frac{b}{0}}, \mathsf{fma}\left(8, {\left(\frac{c}{b}\right)}^{3}, b \cdot 0\right)\right)\right), {a}^{4} \cdot \mathsf{fma}\left(-2, \frac{c}{\frac{b}{0}}, \mathsf{fma}\left(-2, \frac{c \cdot c}{\frac{{b}^{3}}{0}}, \mathsf{fma}\left(-0.5, b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{5}}, b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20 - \frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right)\right)\right)\right)\right)\right)\right)}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.025:\\ \;\;\;\;\frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{{b}^{3} + {\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{1.5}}}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + \left(b \cdot b + b \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, \left(b \cdot c\right) \cdot -6, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(-2, \frac{c \cdot c}{b}, \mathsf{fma}\left(8, \frac{c \cdot c}{b}, b \cdot 0\right)\right), \mathsf{fma}\left({a}^{3}, \mathsf{fma}\left(-4, {\left(\frac{c}{b}\right)}^{3}, \mathsf{fma}\left(-2, \frac{c}{\frac{b}{0}}, \mathsf{fma}\left(8, {\left(\frac{c}{b}\right)}^{3}, b \cdot 0\right)\right)\right), {a}^{4} \cdot \mathsf{fma}\left(-2, \frac{c}{\frac{b}{0}}, \mathsf{fma}\left(-2, \frac{c \cdot c}{\frac{{b}^{3}}{0}}, \mathsf{fma}\left(-0.5, b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20\right), \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{5}}, b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 20 - \frac{{c}^{4}}{{b}^{6}} \cdot 20\right)\right)\right)\right)\right)\right)\right)\right)}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + \left(b \cdot b + b \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 2: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(a \cdot c\right) \cdot 4\\ t_1 := \mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\\ \mathbf{if}\;b \leq 0.0305:\\ \;\;\;\;\frac{\frac{\frac{{t_1}^{3} - {b}^{6}}{{b}^{3} + {t_1}^{1.5}}}{t_0 + \left(b \cdot b + b \cdot \sqrt{t_0}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \left(-5 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4}}} - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* a c) 4.0))) (t_1 (fma b b (* -4.0 (* a c)))))
   (if (<= b 0.0305)
     (/
      (/
       (/ (- (pow t_1 3.0) (pow b 6.0)) (+ (pow b 3.0) (pow t_1 1.5)))
       (+ t_0 (+ (* b b) (* b (sqrt t_0)))))
      (* a 2.0))
     (fma
      -2.0
      (* (/ (* a a) (pow b 5.0)) (pow c 3.0))
      (-
       (-
        (* -5.0 (/ (pow a 3.0) (/ (pow b 7.0) (pow c 4.0))))
        (* (* c c) (/ a (pow b 3.0))))
       (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - ((a * c) * 4.0);
	double t_1 = fma(b, b, (-4.0 * (a * c)));
	double tmp;
	if (b <= 0.0305) {
		tmp = (((pow(t_1, 3.0) - pow(b, 6.0)) / (pow(b, 3.0) + pow(t_1, 1.5))) / (t_0 + ((b * b) + (b * sqrt(t_0))))) / (a * 2.0);
	} else {
		tmp = fma(-2.0, (((a * a) / pow(b, 5.0)) * pow(c, 3.0)), (((-5.0 * (pow(a, 3.0) / (pow(b, 7.0) / pow(c, 4.0)))) - ((c * c) * (a / pow(b, 3.0)))) - (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))
	t_1 = fma(b, b, Float64(-4.0 * Float64(a * c)))
	tmp = 0.0
	if (b <= 0.0305)
		tmp = Float64(Float64(Float64(Float64((t_1 ^ 3.0) - (b ^ 6.0)) / Float64((b ^ 3.0) + (t_1 ^ 1.5))) / Float64(t_0 + Float64(Float64(b * b) + Float64(b * sqrt(t_0))))) / Float64(a * 2.0));
	else
		tmp = fma(-2.0, Float64(Float64(Float64(a * a) / (b ^ 5.0)) * (c ^ 3.0)), Float64(Float64(Float64(-5.0 * Float64((a ^ 3.0) / Float64((b ^ 7.0) / (c ^ 4.0)))) - Float64(Float64(c * c) * Float64(a / (b ^ 3.0)))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0305], N[(N[(N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] + N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(N[(b * b), $MachinePrecision] + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-5.0 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.030499999999999999

   1. Initial program 89.1%

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

   3. Simplified 89.1%

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

      1. *-commutative 89.1%

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

      2. metadata-eval 89.1%

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

      3. distribute-lft-neg-in 89.1%

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

      4. distribute-rgt-neg-in 89.1%

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

      5. *-commutative 89.1%

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

      6. fma-neg 89.1%

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

      7. associate-*l* 89.1%

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

   5. Applied egg-rr 89.1%

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

      1. flip3-- 88.9%

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

      2. add-sqr-sqrt 89.1%

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

   7. Applied egg-rr 89.1%

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

      1. flip-- 88.9%

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

   9. Applied egg-rr 89.4%

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} \cdot {\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. pow-sqr 89.3%

          \[\leadsto \frac{\frac{\frac{\color{blue}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{\left(2 \cdot 1.5\right)}} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       2. fma-def 90.3%

          \[\leadsto \frac{\frac{\frac{{\color{blue}{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}}^{\left(2 \cdot 1.5\right)} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       3. *-commutative 90.3%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)\right)}^{\left(2 \cdot 1.5\right)} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       4. metadata-eval 90.3%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{\color{blue}{3}} - {b}^{3} \cdot {b}^{3}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       5. pow-sqr 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - \color{blue}{{b}^{\left(2 \cdot 3\right)}}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       6. metadata-eval 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{\color{blue}{6}}}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       7. +-commutative 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{\color{blue}{{b}^{3} + {\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5}}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       8. fma-def 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{{b}^{3} + {\color{blue}{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}}^{1.5}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       9. *-commutative 90.0%

          \[\leadsto \frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{{b}^{3} + {\left(\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)\right)}^{1.5}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

   11. Simplified 90.0%

       \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{{b}^{3} + {\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{1.5}}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

   if 0.030499999999999999 < b

   1. Initial program 52.8%

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

   2. Taylor expanded in a around 0 93.2%

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

   4. Simplified 93.2%

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

   5. Taylor expanded in c around 0 93.2%

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

      1. associate-/l* 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0305:\\ \;\;\;\;\frac{\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{3} - {b}^{6}}{{b}^{3} + {\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{1.5}}}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + \left(b \cdot b + b \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \left(-5 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4}}} - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 3: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(a \cdot c\right) \cdot 4\\ \mathbf{if}\;b \leq 0.022:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{1.5} - {b}^{3}}{t_0 + \left(b \cdot b + b \cdot \sqrt{t_0}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \left(-5 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4}}} - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* a c) 4.0))))
   (if (<= b 0.022)
     (/
      (/
       (- (pow (fma b b (* -4.0 (* a c))) 1.5) (pow b 3.0))
       (+ t_0 (+ (* b b) (* b (sqrt t_0)))))
      (* a 2.0))
     (fma
      -2.0
      (* (/ (* a a) (pow b 5.0)) (pow c 3.0))
      (-
       (-
        (* -5.0 (/ (pow a 3.0) (/ (pow b 7.0) (pow c 4.0))))
        (* (* c c) (/ a (pow b 3.0))))
       (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - ((a * c) * 4.0);
	double tmp;
	if (b <= 0.022) {
		tmp = ((pow(fma(b, b, (-4.0 * (a * c))), 1.5) - pow(b, 3.0)) / (t_0 + ((b * b) + (b * sqrt(t_0))))) / (a * 2.0);
	} else {
		tmp = fma(-2.0, (((a * a) / pow(b, 5.0)) * pow(c, 3.0)), (((-5.0 * (pow(a, 3.0) / (pow(b, 7.0) / pow(c, 4.0)))) - ((c * c) * (a / pow(b, 3.0)))) - (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))
	tmp = 0.0
	if (b <= 0.022)
		tmp = Float64(Float64(Float64((fma(b, b, Float64(-4.0 * Float64(a * c))) ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + Float64(Float64(b * b) + Float64(b * sqrt(t_0))))) / Float64(a * 2.0));
	else
		tmp = fma(-2.0, Float64(Float64(Float64(a * a) / (b ^ 5.0)) * (c ^ 3.0)), Float64(Float64(Float64(-5.0 * Float64((a ^ 3.0) / Float64((b ^ 7.0) / (c ^ 4.0)))) - Float64(Float64(c * c) * Float64(a / (b ^ 3.0)))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.022], N[(N[(N[(N[Power[N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(N[(b * b), $MachinePrecision] + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-5.0 * N[(N[Power[a, 3.0], $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.021999999999999999

   1. Initial program 89.1%

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

   3. Simplified 89.1%

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

      1. *-commutative 89.1%

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

      2. metadata-eval 89.1%

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

      3. distribute-lft-neg-in 89.1%

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

      4. distribute-rgt-neg-in 89.1%

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

      5. *-commutative 89.1%

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

      6. fma-neg 89.1%

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

      7. associate-*l* 89.1%

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

   5. Applied egg-rr 89.1%

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

      1. flip3-- 88.9%

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

      2. add-sqr-sqrt 89.1%

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

   7. Applied egg-rr 89.1%

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

      1. sub-neg 89.1%

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

      2. sqrt-pow2 89.3%

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

      3. cancel-sign-sub-inv 89.3%

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

      4. metadata-eval 89.3%

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

      5. *-commutative 89.3%

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

      6. metadata-eval 89.3%

         \[\leadsto \frac{\frac{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{\color{blue}{1.5}} + \left(-{b}^{3}\right)}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

   9. Applied egg-rr 89.3%

      \[\leadsto \frac{\frac{\color{blue}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} + \left(-{b}^{3}\right)}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. sub-neg 89.3%

          \[\leadsto \frac{\frac{\color{blue}{{\left(b \cdot b + -4 \cdot \left(c \cdot a\right)\right)}^{1.5} - {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       2. fma-def 90.0%

          \[\leadsto \frac{\frac{{\color{blue}{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)\right)}}^{1.5} - {b}^{3}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

       3. *-commutative 90.0%

          \[\leadsto \frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)\right)}^{1.5} - {b}^{3}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

   11. Simplified 90.0%

       \[\leadsto \frac{\frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{1.5} - {b}^{3}}}{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot b\right)}}{a \cdot 2} \]

   if 0.021999999999999999 < b

   1. Initial program 52.8%

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

   2. Taylor expanded in a around 0 93.2%

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

   4. Simplified 93.2%

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

   5. Taylor expanded in c around 0 93.2%

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

      1. associate-/l* 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.022:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{1.5} - {b}^{3}}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + \left(b \cdot b + b \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}, \left(-5 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4}}} - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 4: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(a \cdot c\right) \cdot 4\\ \mathbf{if}\;b \leq 0.178:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(a \cdot a\right) \cdot -2}{\frac{{b}^{5}}{{c}^{3}}} - \frac{c}{b}\right) - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* a c) 4.0))))
   (if (<= b 0.178)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* a 2.0))
     (-
      (- (/ (* (* a a) -2.0) (/ (pow b 5.0) (pow c 3.0))) (/ c b))
      (* (* c c) (/ a (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - ((a * c) * 4.0);
	double tmp;
	if (b <= 0.178) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((((a * a) * -2.0) / (pow(b, 5.0) / pow(c, 3.0))) - (c / b)) - ((c * c) * (a / pow(b, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - ((a * c) * 4.0d0)
    if (b <= 0.178d0) then
        tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (a * 2.0d0)
    else
        tmp = ((((a * a) * (-2.0d0)) / ((b ** 5.0d0) / (c ** 3.0d0))) - (c / b)) - ((c * c) * (a / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * b) - ((a * c) * 4.0);
	double tmp;
	if (b <= 0.178) {
		tmp = ((t_0 - (b * b)) / (b + Math.sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((((a * a) * -2.0) / (Math.pow(b, 5.0) / Math.pow(c, 3.0))) - (c / b)) - ((c * c) * (a / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) - ((a * c) * 4.0)
	tmp = 0
	if b <= 0.178:
		tmp = ((t_0 - (b * b)) / (b + math.sqrt(t_0))) / (a * 2.0)
	else:
		tmp = ((((a * a) * -2.0) / (math.pow(b, 5.0) / math.pow(c, 3.0))) - (c / b)) - ((c * c) * (a / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))
	tmp = 0.0
	if (b <= 0.178)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * -2.0) / Float64((b ^ 5.0) / (c ^ 3.0))) - Float64(c / b)) - Float64(Float64(c * c) * Float64(a / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) - ((a * c) * 4.0);
	tmp = 0.0;
	if (b <= 0.178)
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (a * 2.0);
	else
		tmp = ((((a * a) * -2.0) / ((b ^ 5.0) / (c ^ 3.0))) - (c / b)) - ((c * c) * (a / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.178], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.17799999999999999

   1. Initial program 86.6%

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

   3. Simplified 86.6%

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

      1. *-commutative 86.6%

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

      2. metadata-eval 86.6%

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

      3. distribute-lft-neg-in 86.6%

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

      4. distribute-rgt-neg-in 86.6%

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

      5. *-commutative 86.6%

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

      6. fma-neg 86.6%

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

      7. associate-*l* 86.6%

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

   5. Applied egg-rr 86.6%

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

      1. flip-- 86.6%

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

      2. add-sqr-sqrt 88.0%

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

   7. Applied egg-rr 88.0%

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

   if 0.17799999999999999 < b

   1. Initial program 52.4%

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

   2. Taylor expanded in b around inf 91.0%

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

      1. associate-+r+ 91.0%

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

      2. mul-1-neg 91.0%

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

      3. unsub-neg 91.0%

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

      4. mul-1-neg 91.0%

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

      5. unsub-neg 91.0%

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

      6. associate-/l* 91.0%

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

      7. associate-*r/ 91.0%

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

      8. unpow2 91.0%

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

      9. associate-/l* 91.0%

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

      10. associate-/r/ 91.0%

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

      11. unpow2 91.0%

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

   4. Simplified 91.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.178:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(a \cdot a\right) \cdot -2}{\frac{{b}^{5}}{{c}^{3}}} - \frac{c}{b}\right) - \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}\\ \end{array} \]

Alternative 5: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot 4\\ t_1 := b \cdot b - t_0\\ \mathbf{if}\;b \leq 3.715:\\ \;\;\;\;\frac{\frac{t_1 - b \cdot b}{b + \sqrt{t_1}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) 4.0)) (t_1 (- (* b b) t_0)))
   (if (<= b 3.715)
     (/ (/ (- t_1 (* b b)) (+ b (sqrt t_1))) (* a 2.0))
     (/ (/ t_0 (- (- b) (+ b (* c (/ (* a -2.0) b))))) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * 4.0;
	double t_1 = (b * b) - t_0;
	double tmp;
	if (b <= 3.715) {
		tmp = ((t_1 - (b * b)) / (b + sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = (t_0 / (-b - (b + (c * ((a * -2.0) / b))))) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a * c) * 4.0d0
    t_1 = (b * b) - t_0
    if (b <= 3.715d0) then
        tmp = ((t_1 - (b * b)) / (b + sqrt(t_1))) / (a * 2.0d0)
    else
        tmp = (t_0 / (-b - (b + (c * ((a * (-2.0d0)) / b))))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) * 4.0;
	double t_1 = (b * b) - t_0;
	double tmp;
	if (b <= 3.715) {
		tmp = ((t_1 - (b * b)) / (b + Math.sqrt(t_1))) / (a * 2.0);
	} else {
		tmp = (t_0 / (-b - (b + (c * ((a * -2.0) / b))))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (a * c) * 4.0
	t_1 = (b * b) - t_0
	tmp = 0
	if b <= 3.715:
		tmp = ((t_1 - (b * b)) / (b + math.sqrt(t_1))) / (a * 2.0)
	else:
		tmp = (t_0 / (-b - (b + (c * ((a * -2.0) / b))))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * 4.0)
	t_1 = Float64(Float64(b * b) - t_0)
	tmp = 0.0
	if (b <= 3.715)
		tmp = Float64(Float64(Float64(t_1 - Float64(b * b)) / Float64(b + sqrt(t_1))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(-b) - Float64(b + Float64(c * Float64(Float64(a * -2.0) / b))))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (a * c) * 4.0;
	t_1 = (b * b) - t_0;
	tmp = 0.0;
	if (b <= 3.715)
		tmp = ((t_1 - (b * b)) / (b + sqrt(t_1))) / (a * 2.0);
	else
		tmp = (t_0 / (-b - (b + (c * ((a * -2.0) / b))))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[b, 3.715], N[(N[(N[(t$95$1 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[((-b) - N[(b + N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7149999999999999

   1. Initial program 78.4%

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

   3. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. metadata-eval 78.5%

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

      3. distribute-lft-neg-in 78.5%

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

      4. distribute-rgt-neg-in 78.5%

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

      5. *-commutative 78.5%

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

      6. fma-neg 78.4%

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

      7. associate-*l* 78.4%

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

   5. Applied egg-rr 78.4%

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

      1. flip-- 78.7%

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

      2. add-sqr-sqrt 80.5%

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

   7. Applied egg-rr 80.5%

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

   if 3.7149999999999999 < b

   1. Initial program 49.9%

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

   2. Taylor expanded in b around inf 35.9%

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

      1. associate-/l* 35.9%

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

   4. Simplified 35.9%

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

      1. flip-+ 35.9%

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

      2. associate-*r/ 35.9%

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

      3. associate-*r/ 35.9%

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

      4. associate-*r/ 35.9%

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

   6. Applied egg-rr 35.9%

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

      1. sqr-neg 35.9%

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

      2. associate-/r/ 35.9%

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

      3. *-commutative 35.9%

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

      4. associate-/r/ 35.9%

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

      5. *-commutative 35.9%

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

      6. associate-/r/ 35.9%

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

      7. *-commutative 35.9%

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

   8. Simplified 35.9%

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

   9. Taylor expanded in b around inf 87.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.715:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 6: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot 4\\ \mathbf{if}\;b \leq 0.185:\\ \;\;\;\;\frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) 4.0)))
   (if (<= b 0.185)
     (/ (- (sqrt (- (* b b) t_0)) b) (* a 2.0))
     (/ (/ t_0 (- (- b) (+ b (* c (/ (* a -2.0) b))))) (* a 2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * 4.0;
	double tmp;
	if (b <= 0.185) {
		tmp = (sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	} else {
		tmp = (t_0 / (-b - (b + (c * ((a * -2.0) / b))))) / (a * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * c) * 4.0d0
    if (b <= 0.185d0) then
        tmp = (sqrt(((b * b) - t_0)) - b) / (a * 2.0d0)
    else
        tmp = (t_0 / (-b - (b + (c * ((a * (-2.0d0)) / b))))) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) * 4.0;
	double tmp;
	if (b <= 0.185) {
		tmp = (Math.sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	} else {
		tmp = (t_0 / (-b - (b + (c * ((a * -2.0) / b))))) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (a * c) * 4.0
	tmp = 0
	if b <= 0.185:
		tmp = (math.sqrt(((b * b) - t_0)) - b) / (a * 2.0)
	else:
		tmp = (t_0 / (-b - (b + (c * ((a * -2.0) / b))))) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * 4.0)
	tmp = 0.0
	if (b <= 0.185)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(-b) - Float64(b + Float64(c * Float64(Float64(a * -2.0) / b))))) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (a * c) * 4.0;
	tmp = 0.0;
	if (b <= 0.185)
		tmp = (sqrt(((b * b) - t_0)) - b) / (a * 2.0);
	else
		tmp = (t_0 / (-b - (b + (c * ((a * -2.0) / b))))) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[b, 0.185], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[((-b) - N[(b + N[(c * N[(N[(a * -2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.185

   1. Initial program 86.3%

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

   3. Simplified 86.3%

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

      1. *-commutative 86.3%

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

      2. metadata-eval 86.3%

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

      3. distribute-lft-neg-in 86.3%

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

      4. distribute-rgt-neg-in 86.3%

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

      5. *-commutative 86.3%

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

      6. fma-neg 86.3%

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

      7. associate-*l* 86.3%

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

   5. Applied egg-rr 86.3%

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

   if 0.185 < b

   1. Initial program 52.2%

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

   2. Taylor expanded in b around inf 36.6%

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

      1. associate-/l* 36.6%

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

   4. Simplified 36.6%

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

      1. flip-+ 36.5%

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

      2. associate-*r/ 36.5%

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

      3. associate-*r/ 36.5%

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

      4. associate-*r/ 36.5%

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

   6. Applied egg-rr 36.5%

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

      1. sqr-neg 36.5%

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

      2. associate-/r/ 36.5%

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

      3. *-commutative 36.5%

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

      4. associate-/r/ 36.5%

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

      5. *-commutative 36.5%

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

      6. associate-/r/ 36.5%

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

      7. *-commutative 36.5%

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

   8. Simplified 36.5%

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

   9. Taylor expanded in b around inf 86.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.185:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \left(b + c \cdot \frac{a \cdot -2}{b}\right)}}{a \cdot 2}\\ \end{array} \]

Alternative 7: 82.3% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.0%

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

2. Taylor expanded in b around inf 36.5%

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

   1. associate-/l* 36.5%

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

4. Simplified 36.5%

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

   1. flip-+ 36.5%

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

   2. associate-*r/ 36.5%

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

   3. associate-*r/ 36.5%

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

   4. associate-*r/ 36.5%

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

6. Applied egg-rr 36.5%

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

   1. sqr-neg 36.5%

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

   2. associate-/r/ 36.5%

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

   3. *-commutative 36.5%

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

   4. associate-/r/ 36.5%

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

   5. *-commutative 36.5%

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

   6. associate-/r/ 36.5%

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

   7. *-commutative 36.5%

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

8. Simplified 36.5%

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

9. Taylor expanded in b around inf 83.3%

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

10. Final simplification 83.3%

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

Alternative 8: 64.5% 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 55.0%

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

2. Taylor expanded in b around inf 65.3%

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

   1. mul-1-neg 65.3%

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

   2. distribute-neg-frac 65.3%

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

4. Simplified 65.3%

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

5. Final simplification 65.3%

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

Alternative 9: 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 55.0%

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

2. Taylor expanded in b around -inf 11.5%

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

   1. +-commutative 11.5%

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

   2. mul-1-neg 11.5%

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

   3. unsub-neg 11.5%

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

4. Simplified 11.5%

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

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 2023293 
(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)))