Quadratic roots, narrow range

Percentage Accurate: 55.4% → 91.4%

Time: 27.9s

Alternatives: 14

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {c}^{4} \cdot {a}^{4}\\ t_1 := \frac{4 \cdot t_0 + t_0 \cdot 16}{{b}^{5}}\\ t_2 := a \cdot \left(c \cdot 0\right)\\ t_3 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{t_2}{{b}^{3}}, \mathsf{fma}\left(-0.5, t_1, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{t_2}{{b}^{5}}, \mathsf{fma}\left(\frac{c \cdot c}{b} \cdot 2, a \cdot a, 4 \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{5}} - {a}^{3} \cdot \left({\left(\frac{c}{b}\right)}^{3} \cdot -12\right)\right)\right)\right)\right) - t_1\right)\right)\right)\right)}{t_3 + b \cdot \left(b + \sqrt{t_3}\right)}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (pow c 4.0) (pow a 4.0)))
        (t_1 (/ (+ (* 4.0 t_0) (* t_0 16.0)) (pow b 5.0)))
        (t_2 (* a (* c 0.0)))
        (t_3 (fma b b (* a (* c -4.0)))))
   (/
    (/
     (fma
      4.0
      (/ (* c c) (/ b (* a a)))
      (fma
       -2.0
       (/ t_2 (pow b 3.0))
       (fma
        -0.5
        t_1
        (fma
         (* (* c a) -6.0)
         b
         (-
          (fma
           32.0
           (/ (pow c 4.0) (/ (pow b 5.0) (pow a 4.0)))
           (fma
            -8.0
            (/ (pow c 3.0) (/ (pow b 3.0) (pow a 3.0)))
            (fma
             -2.0
             (/ t_2 (pow b 5.0))
             (fma
              (* (/ (* c c) b) 2.0)
              (* a a)
              (-
               (* 4.0 (/ (pow (* c a) 4.0) (pow b 5.0)))
               (* (pow a 3.0) (* (pow (/ c b) 3.0) -12.0)))))))
          t_1)))))
     (+ t_3 (* b (+ b (sqrt t_3)))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = pow(c, 4.0) * pow(a, 4.0);
	double t_1 = ((4.0 * t_0) + (t_0 * 16.0)) / pow(b, 5.0);
	double t_2 = a * (c * 0.0);
	double t_3 = fma(b, b, (a * (c * -4.0)));
	return (fma(4.0, ((c * c) / (b / (a * a))), fma(-2.0, (t_2 / pow(b, 3.0)), fma(-0.5, t_1, fma(((c * a) * -6.0), b, (fma(32.0, (pow(c, 4.0) / (pow(b, 5.0) / pow(a, 4.0))), fma(-8.0, (pow(c, 3.0) / (pow(b, 3.0) / pow(a, 3.0))), fma(-2.0, (t_2 / pow(b, 5.0)), fma((((c * c) / b) * 2.0), (a * a), ((4.0 * (pow((c * a), 4.0) / pow(b, 5.0))) - (pow(a, 3.0) * (pow((c / b), 3.0) * -12.0))))))) - t_1))))) / (t_3 + (b * (b + sqrt(t_3))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	t_0 = Float64((c ^ 4.0) * (a ^ 4.0))
	t_1 = Float64(Float64(Float64(4.0 * t_0) + Float64(t_0 * 16.0)) / (b ^ 5.0))
	t_2 = Float64(a * Float64(c * 0.0))
	t_3 = fma(b, b, Float64(a * Float64(c * -4.0)))
	return Float64(Float64(fma(4.0, Float64(Float64(c * c) / Float64(b / Float64(a * a))), fma(-2.0, Float64(t_2 / (b ^ 3.0)), fma(-0.5, t_1, fma(Float64(Float64(c * a) * -6.0), b, Float64(fma(32.0, Float64((c ^ 4.0) / Float64((b ^ 5.0) / (a ^ 4.0))), fma(-8.0, Float64((c ^ 3.0) / Float64((b ^ 3.0) / (a ^ 3.0))), fma(-2.0, Float64(t_2 / (b ^ 5.0)), fma(Float64(Float64(Float64(c * c) / b) * 2.0), Float64(a * a), Float64(Float64(4.0 * Float64((Float64(c * a) ^ 4.0) / (b ^ 5.0))) - Float64((a ^ 3.0) * Float64((Float64(c / b) ^ 3.0) * -12.0))))))) - t_1))))) / Float64(t_3 + Float64(b * Float64(b + sqrt(t_3))))) / Float64(a * 2.0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(4.0 * t$95$0), $MachinePrecision] + N[(t$95$0 * 16.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(4.0 * N[(N[(c * c), $MachinePrecision] / N[(b / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(t$95$2 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * t$95$1 + N[(N[(N[(c * a), $MachinePrecision] * -6.0), $MachinePrecision] * b + N[(N[(32.0 * N[(N[Power[c, 4.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-8.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(t$95$2 / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * 2.0), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(4.0 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[Power[N[(c / b), $MachinePrecision], 3.0], $MachinePrecision] * -12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 + N[(b * N[(b + N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 55.4%

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

   1. *-commutative 55.4%

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

   2. +-commutative 55.4%

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

   3. unsub-neg 55.4%

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

   4. fma-neg 55.3%

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

   5. associate-*l* 55.3%

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

   6. *-commutative 55.3%

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

   7. distribute-rgt-neg-in 55.3%

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

   8. metadata-eval 55.3%

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

3. Simplified 55.3%

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

   1. fma-udef 55.4%

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

   2. associate-*l* 55.4%

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

5. Applied egg-rr 55.4%

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

   1. flip3-- 55.4%

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

   2. fma-def 55.4%

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

   3. add-sqr-sqrt 55.4%

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

   4. fma-def 55.4%

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

   5. fma-def 55.4%

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

7. Applied egg-rr 55.4%

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

   1. distribute-rgt-out 55.4%

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

9. Simplified 55.4%

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

10. Taylor expanded in b around inf 91.6%

    \[\leadsto \frac{\frac{\color{blue}{4 \cdot \frac{{c}^{2} \cdot {a}^{2}}{b} + \left(-2 \cdot \frac{c \cdot \left(a \cdot \left(-4 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 4 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)\right)}{{b}^{3}} + \left(-0.5 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}} + \left(\left(-4 \cdot \left(c \cdot a\right) + -2 \cdot \left(c \cdot a\right)\right) \cdot b + \left(-1 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}} + \left(32 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}} + \left(-8 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{3}} + \left(-2 \cdot \frac{c \cdot \left(\left(-8 \cdot \left({c}^{3} \cdot {a}^{3}\right) + 8 \cdot \left({c}^{3} \cdot {a}^{3}\right)\right) \cdot a\right)}{{b}^{5}} + \left(-4 \cdot \frac{{c}^{2} \cdot {a}^{2}}{b} + \left(16 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{3}} + \left(8 \cdot \frac{{c}^{2} \cdot {a}^{2}}{b} + \left(-4 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{3}} + \left(-2 \cdot \frac{{c}^{2} \cdot \left({a}^{2} \cdot \left(-4 \cdot \left({c}^{2} \cdot {a}^{2}\right) + 4 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)\right)}{{b}^{5}} + \left(-2 \cdot \frac{{c}^{2} \cdot {a}^{2}}{b} + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

12. Simplified 91.7%

    \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \mathsf{fma}\left(-4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(16, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(8, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-4, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \frac{0}{{b}^{5}} + \mathsf{fma}\left(-2, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

13. Taylor expanded in a around -inf 91.7%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \color{blue}{\left(-2 \cdot \frac{{c}^{2}}{b} + \left(8 \cdot \frac{{c}^{2}}{b} + -4 \cdot \frac{{c}^{2}}{b}\right)\right) \cdot {a}^{2} + \left(-1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right) + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)}\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]
14. Step-by-step derivation

    1. fma-def 91.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \color{blue}{\mathsf{fma}\left(-2 \cdot \frac{{c}^{2}}{b} + \left(8 \cdot \frac{{c}^{2}}{b} + -4 \cdot \frac{{c}^{2}}{b}\right), {a}^{2}, -1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right) + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)}\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

    2. *-commutative 91.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \mathsf{fma}\left(\color{blue}{\frac{{c}^{2}}{b} \cdot -2} + \left(8 \cdot \frac{{c}^{2}}{b} + -4 \cdot \frac{{c}^{2}}{b}\right), {a}^{2}, -1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right) + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

    3. distribute-rgt-out 91.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \mathsf{fma}\left(\frac{{c}^{2}}{b} \cdot -2 + \color{blue}{\frac{{c}^{2}}{b} \cdot \left(8 + -4\right)}, {a}^{2}, -1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right) + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

    4. metadata-eval 91.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \mathsf{fma}\left(\frac{{c}^{2}}{b} \cdot -2 + \frac{{c}^{2}}{b} \cdot \color{blue}{4}, {a}^{2}, -1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right) + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

    5. distribute-lft-out 91.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \mathsf{fma}\left(\color{blue}{\frac{{c}^{2}}{b} \cdot \left(-2 + 4\right)}, {a}^{2}, -1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right) + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

    6. unpow2 91.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \mathsf{fma}\left(\frac{\color{blue}{c \cdot c}}{b} \cdot \left(-2 + 4\right), {a}^{2}, -1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right) + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

    7. metadata-eval 91.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \mathsf{fma}\left(\frac{c \cdot c}{b} \cdot \color{blue}{2}, {a}^{2}, -1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right) + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

    8. unpow2 91.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \mathsf{fma}\left(\frac{c \cdot c}{b} \cdot 2, \color{blue}{a \cdot a}, -1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right) + 4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}}\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

    9. +-commutative 91.7%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \mathsf{fma}\left(\frac{c \cdot c}{b} \cdot 2, a \cdot a, \color{blue}{4 \cdot \frac{{c}^{4} \cdot {a}^{4}}{{b}^{5}} + -1 \cdot \left({a}^{3} \cdot \left(-16 \cdot \frac{{c}^{3}}{{b}^{3}} + 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right)}\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

15. Simplified 91.7%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{\left(c \cdot 0\right) \cdot a}{{b}^{5}}, \color{blue}{\mathsf{fma}\left(\frac{c \cdot c}{b} \cdot 2, a \cdot a, 4 \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{5}} - {a}^{3} \cdot \left({\left(\frac{c}{b}\right)}^{3} \cdot -12\right)\right)}\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

16. Final simplification 91.7%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(4, \frac{c \cdot c}{\frac{b}{a \cdot a}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot 0\right)}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + \left({c}^{4} \cdot {a}^{4}\right) \cdot 16}{{b}^{5}}, \mathsf{fma}\left(\left(c \cdot a\right) \cdot -6, b, \mathsf{fma}\left(32, \frac{{c}^{4}}{\frac{{b}^{5}}{{a}^{4}}}, \mathsf{fma}\left(-8, \frac{{c}^{3}}{\frac{{b}^{3}}{{a}^{3}}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot 0\right)}{{b}^{5}}, \mathsf{fma}\left(\frac{c \cdot c}{b} \cdot 2, a \cdot a, 4 \cdot \frac{{\left(c \cdot a\right)}^{4}}{{b}^{5}} - {a}^{3} \cdot \left({\left(\frac{c}{b}\right)}^{3} \cdot -12\right)\right)\right)\right)\right) - \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + \left({c}^{4} \cdot {a}^{4}\right) \cdot 16}{{b}^{5}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

Alternative 2: 91.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\frac{b}{0}}\\ t_1 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ t_2 := \frac{{c}^{4}}{{b}^{6}}\\ t_3 := \mathsf{fma}\left(16, t_2, 4 \cdot t_2\right)\\ \frac{\frac{\mathsf{fma}\left(\frac{c \cdot c}{b} \cdot 6, a \cdot a, \mathsf{fma}\left(-6 \cdot \left(c \cdot b\right), a, \mathsf{fma}\left({a}^{4}, \mathsf{fma}\left(b, t_3 - t_3, \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{5}}, \mathsf{fma}\left(-2, t_0, \mathsf{fma}\left(-0.5, b \cdot t_3, -2 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{0}}\right)\right)\right)\right), {a}^{3} \cdot \mathsf{fma}\left(-2, t_0, 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right)\right)\right)}{t_1 + b \cdot \left(b + \sqrt{t_1}\right)}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (/ b 0.0)))
        (t_1 (fma b b (* a (* c -4.0))))
        (t_2 (/ (pow c 4.0) (pow b 6.0)))
        (t_3 (fma 16.0 t_2 (* 4.0 t_2))))
   (/
    (/
     (fma
      (* (/ (* c c) b) 6.0)
      (* a a)
      (fma
       (* -6.0 (* c b))
       a
       (fma
        (pow a 4.0)
        (fma
         b
         (- t_3 t_3)
         (fma
          16.0
          (/ (pow c 4.0) (pow b 5.0))
          (fma
           -2.0
           t_0
           (fma -0.5 (* b t_3) (* -2.0 (/ (* c c) (/ (pow b 3.0) 0.0)))))))
        (* (pow a 3.0) (fma -2.0 t_0 (* 4.0 (/ (pow c 3.0) (pow b 3.0))))))))
     (+ t_1 (* b (+ b (sqrt t_1)))))
    (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = c / (b / 0.0);
	double t_1 = fma(b, b, (a * (c * -4.0)));
	double t_2 = pow(c, 4.0) / pow(b, 6.0);
	double t_3 = fma(16.0, t_2, (4.0 * t_2));
	return (fma((((c * c) / b) * 6.0), (a * a), fma((-6.0 * (c * b)), a, fma(pow(a, 4.0), fma(b, (t_3 - t_3), fma(16.0, (pow(c, 4.0) / pow(b, 5.0)), fma(-2.0, t_0, fma(-0.5, (b * t_3), (-2.0 * ((c * c) / (pow(b, 3.0) / 0.0))))))), (pow(a, 3.0) * fma(-2.0, t_0, (4.0 * (pow(c, 3.0) / pow(b, 3.0)))))))) / (t_1 + (b * (b + sqrt(t_1))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	t_0 = Float64(c / Float64(b / 0.0))
	t_1 = fma(b, b, Float64(a * Float64(c * -4.0)))
	t_2 = Float64((c ^ 4.0) / (b ^ 6.0))
	t_3 = fma(16.0, t_2, Float64(4.0 * t_2))
	return Float64(Float64(fma(Float64(Float64(Float64(c * c) / b) * 6.0), Float64(a * a), fma(Float64(-6.0 * Float64(c * b)), a, fma((a ^ 4.0), fma(b, Float64(t_3 - t_3), fma(16.0, Float64((c ^ 4.0) / (b ^ 5.0)), fma(-2.0, t_0, fma(-0.5, Float64(b * t_3), Float64(-2.0 * Float64(Float64(c * c) / Float64((b ^ 3.0) / 0.0))))))), Float64((a ^ 3.0) * fma(-2.0, t_0, Float64(4.0 * Float64((c ^ 3.0) / (b ^ 3.0)))))))) / Float64(t_1 + Float64(b * Float64(b + sqrt(t_1))))) / Float64(a * 2.0))
end

Wolfram

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

Derivation

1. Initial program 55.4%

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

   1. *-commutative 55.4%

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

   2. +-commutative 55.4%

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

   3. unsub-neg 55.4%

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

   4. fma-neg 55.3%

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

   5. associate-*l* 55.3%

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

   6. *-commutative 55.3%

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

   7. distribute-rgt-neg-in 55.3%

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

   8. metadata-eval 55.3%

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

3. Simplified 55.3%

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

   1. fma-udef 55.4%

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

   2. associate-*l* 55.4%

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

5. Applied egg-rr 55.4%

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

   1. flip3-- 55.4%

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

   2. fma-def 55.4%

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

   3. add-sqr-sqrt 55.4%

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

   4. fma-def 55.4%

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

   5. fma-def 55.4%

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

7. Applied egg-rr 55.4%

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

   1. distribute-rgt-out 55.4%

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

9. Simplified 55.4%

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

10. Taylor expanded in a around 0 91.6%

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

12. Simplified 91.6%

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

13. Final simplification 91.6%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{c \cdot c}{b} \cdot 6, a \cdot a, \mathsf{fma}\left(-6 \cdot \left(c \cdot b\right), a, \mathsf{fma}\left({a}^{4}, \mathsf{fma}\left(b, \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right) - \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{5}}, \mathsf{fma}\left(-2, \frac{c}{\frac{b}{0}}, \mathsf{fma}\left(-0.5, b \cdot \mathsf{fma}\left(16, \frac{{c}^{4}}{{b}^{6}}, 4 \cdot \frac{{c}^{4}}{{b}^{6}}\right), -2 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{0}}\right)\right)\right)\right), {a}^{3} \cdot \mathsf{fma}\left(-2, \frac{c}{\frac{b}{0}}, 4 \cdot \frac{{c}^{3}}{{b}^{3}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

Alternative 3: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{{t_0}^{1.5} - {b}^{3}}{t_0 + b \cdot \left(b + \sqrt{t_0}\right)} \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{\left(a \cdot a\right) \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -2.6)
     (*
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ t_0 (* b (+ b (sqrt t_0)))))
      (/ 1.0 (* a 2.0)))
     (-
      (- (/ -2.0 (/ (pow b 5.0) (* (* a a) (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 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -2.6) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) / (t_0 + (b * (b + sqrt(t_0))))) * (1.0 / (a * 2.0));
	} else {
		tmp = ((-2.0 / (pow(b, 5.0) / ((a * a) * pow(c, 3.0)))) - (c / b)) - ((c * (c * a)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -2.6)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + Float64(b * Float64(b + sqrt(t_0))))) * Float64(1.0 / Float64(a * 2.0)));
	else
		tmp = Float64(Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64(Float64(a * a) * (c ^ 3.0)))) - Float64(c / b)) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.6], N[(N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(b * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2.60000000000000009

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. +-commutative 84.4%

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

      3. unsub-neg 84.4%

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

      4. fma-neg 84.7%

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

      5. associate-*l* 84.7%

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

      6. *-commutative 84.7%

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

      7. distribute-rgt-neg-in 84.7%

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

      8. metadata-eval 84.7%

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

   3. Simplified 84.7%

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

      1. fma-udef 84.4%

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

      2. associate-*l* 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. flip3-- 84.4%

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

      2. fma-def 84.6%

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

      3. add-sqr-sqrt 84.6%

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

      4. fma-def 84.6%

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

      5. fma-def 84.6%

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

   7. Applied egg-rr 84.6%

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

      1. distribute-rgt-out 84.7%

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

   9. Simplified 84.7%

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

       1. div-inv 84.7%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3} - {b}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)} \cdot \frac{1}{a \cdot 2}} \]

       2. sqrt-pow2 85.1%

          \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{\left(\frac{3}{2}\right)}} - {b}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)} \cdot \frac{1}{a \cdot 2} \]

       3. metadata-eval 85.1%

          \[\leadsto \frac{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{\color{blue}{1.5}} - {b}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)} \cdot \frac{1}{a \cdot 2} \]

   11. Applied egg-rr 85.1%

       \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)} \cdot \frac{1}{a \cdot 2}} \]

   if -2.60000000000000009 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.2%

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

      1. neg-sub0 51.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 51.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 51.2%

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

      4. neg-mul-1 51.2%

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

      5. associate-*l/ 51.2%

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

      6. *-commutative 51.2%

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

      7. associate-/r* 51.2%

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

      8. /-rgt-identity 51.2%

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

      9. metadata-eval 51.2%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in b around inf 90.4%

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

      1. +-commutative 90.4%

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

      2. mul-1-neg 90.4%

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

      3. unsub-neg 90.4%

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

      4. +-commutative 90.4%

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

      5. mul-1-neg 90.4%

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

      6. unsub-neg 90.4%

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

      7. associate-*r/ 90.4%

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

      8. associate-/l* 90.4%

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

      9. unpow2 90.4%

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

      10. unpow2 90.4%

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

      11. associate-*l* 90.4%

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

   6. Simplified 90.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)} \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{\left(a \cdot a\right) \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 4: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{\left({t_0}^{1.5} - {b}^{3}\right) \cdot \frac{1}{t_0 + b \cdot \left(b + \sqrt{t_0}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{\left(a \cdot a\right) \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -2.6)
     (/
      (* (- (pow t_0 1.5) (pow b 3.0)) (/ 1.0 (+ t_0 (* b (+ b (sqrt t_0))))))
      (* a 2.0))
     (-
      (- (/ -2.0 (/ (pow b 5.0) (* (* a a) (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 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -2.6) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) * (1.0 / (t_0 + (b * (b + sqrt(t_0)))))) / (a * 2.0);
	} else {
		tmp = ((-2.0 / (pow(b, 5.0) / ((a * a) * pow(c, 3.0)))) - (c / b)) - ((c * (c * a)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -2.6)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) * Float64(1.0 / Float64(t_0 + Float64(b * Float64(b + sqrt(t_0)))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64(Float64(a * a) * (c ^ 3.0)))) - Float64(c / b)) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.6], N[(N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$0 + N[(b * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2.60000000000000009

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. +-commutative 84.4%

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

      3. unsub-neg 84.4%

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

      4. fma-neg 84.7%

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

      5. associate-*l* 84.7%

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

      6. *-commutative 84.7%

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

      7. distribute-rgt-neg-in 84.7%

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

      8. metadata-eval 84.7%

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

   3. Simplified 84.7%

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

      1. fma-udef 84.4%

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

      2. associate-*l* 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. flip3-- 84.4%

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

      2. fma-def 84.6%

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

      3. add-sqr-sqrt 84.6%

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

      4. fma-def 84.6%

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

      5. fma-def 84.6%

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

   7. Applied egg-rr 84.6%

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

      1. distribute-rgt-out 84.7%

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

   9. Simplified 84.7%

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

       1. div-inv 84.7%

          \[\leadsto \frac{\color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}^{3} - {b}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]

       2. sqrt-pow2 85.1%

          \[\leadsto \frac{\left(\color{blue}{{\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{\left(\frac{3}{2}\right)}} - {b}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

       3. metadata-eval 85.1%

          \[\leadsto \frac{\left({\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{\color{blue}{1.5}} - {b}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2} \]

   11. Applied egg-rr 85.1%

       \[\leadsto \frac{\color{blue}{\left({\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}}{a \cdot 2} \]

   if -2.60000000000000009 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.2%

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

      1. neg-sub0 51.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 51.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 51.2%

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

      4. neg-mul-1 51.2%

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

      5. associate-*l/ 51.2%

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

      6. *-commutative 51.2%

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

      7. associate-/r* 51.2%

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

      8. /-rgt-identity 51.2%

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

      9. metadata-eval 51.2%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in b around inf 90.4%

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

      1. +-commutative 90.4%

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

      2. mul-1-neg 90.4%

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

      3. unsub-neg 90.4%

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

      4. +-commutative 90.4%

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

      5. mul-1-neg 90.4%

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

      6. unsub-neg 90.4%

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

      7. associate-*r/ 90.4%

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

      8. associate-/l* 90.4%

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

      9. unpow2 90.4%

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

      10. unpow2 90.4%

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

      11. associate-*l* 90.4%

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

   6. Simplified 90.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{\left({\left(\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}\right) \cdot \frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{\left(a \cdot a\right) \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 5: 90.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

   1. neg-sub0 55.4%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 55.4%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 55.4%

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

   4. neg-mul-1 55.4%

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

   5. associate-*l/ 55.4%

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

   6. *-commutative 55.4%

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

   7. associate-/r* 55.4%

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

   8. /-rgt-identity 55.4%

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

   9. metadata-eval 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in a around 0 91.2%

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

6. Simplified 91.2%

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

7. Taylor expanded in c around 0 91.2%

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

   1. associate-/l* 91.2%

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

9. Simplified 91.2%

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

10. Final simplification 91.2%

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

Alternative 6: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{\left(a \cdot a\right) \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -4.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -2.6)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* a 2.0))
     (-
      (- (/ -2.0 (/ (pow b 5.0) (* (* a a) (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 = fma(c, (a * -4.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -2.6) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((-2.0 / (pow(b, 5.0) / ((a * a) * pow(c, 3.0)))) - (c / b)) - ((c * (c * a)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -4.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -2.6)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64(Float64(a * a) * (c ^ 3.0)))) - Float64(c / b)) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.6], 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[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2.60000000000000009

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

      2. +-commutative 84.4%

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

      3. unsub-neg 84.4%

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

      4. fma-neg 84.7%

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

      5. associate-*l* 84.7%

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

      6. *-commutative 84.7%

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

      7. distribute-rgt-neg-in 84.7%

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

      8. metadata-eval 84.7%

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

   3. Simplified 84.7%

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

      1. fma-udef 84.4%

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

      2. associate-*l* 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. flip-- 83.8%

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

      2. add-sqr-sqrt 85.1%

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

      3. fma-def 84.7%

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

      4. fma-def 84.8%

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

   7. Applied egg-rr 84.8%

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

      1. fma-udef 85.1%

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

      2. associate-*r* 85.1%

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

      3. *-commutative 85.1%

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

      4. metadata-eval 85.1%

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

      5. distribute-rgt-neg-in 85.1%

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

      6. associate-*r* 85.1%

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

      7. +-commutative 85.1%

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

      8. distribute-rgt-neg-in 85.1%

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

      9. fma-def 85.2%

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

      10. distribute-rgt-neg-in 85.2%

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

      11. metadata-eval 85.2%

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

      12. +-commutative 85.2%

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

      13. fma-udef 85.1%

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

      14. associate-*r* 85.1%

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

      15. *-commutative 85.1%

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

      16. metadata-eval 85.1%

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

      17. distribute-rgt-neg-in 85.1%

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

      18. associate-*r* 85.1%

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

      19. +-commutative 85.1%

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

   9. Simplified 85.1%

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

   if -2.60000000000000009 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.2%

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

      1. neg-sub0 51.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 51.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 51.2%

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

      4. neg-mul-1 51.2%

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

      5. associate-*l/ 51.2%

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

      6. *-commutative 51.2%

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

      7. associate-/r* 51.2%

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

      8. /-rgt-identity 51.2%

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

      9. metadata-eval 51.2%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in b around inf 90.4%

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

      1. +-commutative 90.4%

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

      2. mul-1-neg 90.4%

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

      3. unsub-neg 90.4%

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

      4. +-commutative 90.4%

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

      5. mul-1-neg 90.4%

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

      6. unsub-neg 90.4%

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

      7. associate-*r/ 90.4%

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

      8. associate-/l* 90.4%

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

      9. unpow2 90.4%

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

      10. unpow2 90.4%

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

      11. associate-*l* 90.4%

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

   6. Simplified 90.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{\left(a \cdot a\right) \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 7: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - t_0}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{\left(a \cdot a\right) \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* c (* 4.0 a)))) (t_1 (sqrt t_0)))
   (if (<= (/ (- t_1 b) (* a 2.0)) -2.6)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) t_1)) (* a 2.0))
     (-
      (- (/ -2.0 (/ (pow b 5.0) (* (* a a) (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) - (c * (4.0 * a));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -2.6) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = ((-2.0 / (pow(b, 5.0) / ((a * a) * 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) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (c * (4.0d0 * a))
    t_1 = sqrt(t_0)
    if (((t_1 - b) / (a * 2.0d0)) <= (-2.6d0)) then
        tmp = (((-b ** 2.0d0) - t_0) / (-b - t_1)) / (a * 2.0d0)
    else
        tmp = (((-2.0d0) / ((b ** 5.0d0) / ((a * a) * (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) - (c * (4.0 * a));
	double t_1 = Math.sqrt(t_0);
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -2.6) {
		tmp = ((Math.pow(-b, 2.0) - t_0) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = ((-2.0 / (Math.pow(b, 5.0) / ((a * a) * 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) - (c * (4.0 * a))
	t_1 = math.sqrt(t_0)
	tmp = 0
	if ((t_1 - b) / (a * 2.0)) <= -2.6:
		tmp = ((math.pow(-b, 2.0) - t_0) / (-b - t_1)) / (a * 2.0)
	else:
		tmp = ((-2.0 / (math.pow(b, 5.0) / ((a * a) * 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(c * Float64(4.0 * a)))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -2.6)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 / Float64((b ^ 5.0) / Float64(Float64(a * a) * (c ^ 3.0)))) - Float64(c / b)) - Float64(Float64(c * Float64(c * a)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) - (c * (4.0 * a));
	t_1 = sqrt(t_0);
	tmp = 0.0;
	if (((t_1 - b) / (a * 2.0)) <= -2.6)
		tmp = (((-b ^ 2.0) - t_0) / (-b - t_1)) / (a * 2.0);
	else
		tmp = ((-2.0 / ((b ^ 5.0) / ((a * a) * (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[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -2.6], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 / N[(N[Power[b, 5.0], $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2.60000000000000009

   1. Initial program 84.4%

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

      1. flip-+ 83.8%

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

      2. pow2 83.8%

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

      3. add-sqr-sqrt 85.1%

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

      4. *-commutative 85.1%

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

      5. *-commutative 85.1%

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

      6. *-commutative 85.1%

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

      7. *-commutative 85.1%

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

   3. Applied egg-rr 85.1%

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

   if -2.60000000000000009 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.2%

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

      1. neg-sub0 51.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 51.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 51.2%

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

      4. neg-mul-1 51.2%

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

      5. associate-*l/ 51.2%

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

      6. *-commutative 51.2%

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

      7. associate-/r* 51.2%

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

      8. /-rgt-identity 51.2%

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

      9. metadata-eval 51.2%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in b around inf 90.4%

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

      1. +-commutative 90.4%

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

      2. mul-1-neg 90.4%

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

      3. unsub-neg 90.4%

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

      4. +-commutative 90.4%

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

      5. mul-1-neg 90.4%

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

      6. unsub-neg 90.4%

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

      7. associate-*r/ 90.4%

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

      8. associate-/l* 90.4%

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

      9. unpow2 90.4%

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

      10. unpow2 90.4%

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

      11. associate-*l* 90.4%

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

   6. Simplified 90.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -2.6:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{\frac{{b}^{5}}{\left(a \cdot a\right) \cdot {c}^{3}}} - \frac{c}{b}\right) - \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\\ \end{array} \]

Alternative 8: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -0.00025:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - t_0}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* c (* 4.0 a)))) (t_1 (sqrt t_0)))
   (if (<= (/ (- t_1 b) (* a 2.0)) -0.00025)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) t_1)) (* a 2.0))
     (- (/ (- c) b) (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - (c * (4.0 * a));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.00025) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((c * c) / (pow(b, 3.0) / a));
	}
	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 = (b * b) - (c * (4.0d0 * a))
    t_1 = sqrt(t_0)
    if (((t_1 - b) / (a * 2.0d0)) <= (-0.00025d0)) then
        tmp = (((-b ** 2.0d0) - t_0) / (-b - t_1)) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((c * c) / ((b ** 3.0d0) / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * b) - (c * (4.0 * a));
	double t_1 = Math.sqrt(t_0);
	double tmp;
	if (((t_1 - b) / (a * 2.0)) <= -0.00025) {
		tmp = ((Math.pow(-b, 2.0) - t_0) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((c * c) / (Math.pow(b, 3.0) / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) - (c * (4.0 * a))
	t_1 = math.sqrt(t_0)
	tmp = 0
	if ((t_1 - b) / (a * 2.0)) <= -0.00025:
		tmp = ((math.pow(-b, 2.0) - t_0) / (-b - t_1)) / (a * 2.0)
	else:
		tmp = (-c / b) - ((c * c) / (math.pow(b, 3.0) / a))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(t_1 - b) / Float64(a * 2.0)) <= -0.00025)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) - (c * (4.0 * a));
	t_1 = sqrt(t_0);
	tmp = 0.0;
	if (((t_1 - b) / (a * 2.0)) <= -0.00025)
		tmp = (((-b ^ 2.0) - t_0) / (-b - t_1)) / (a * 2.0);
	else
		tmp = (-c / b) - ((c * c) / ((b ^ 3.0) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2.5000000000000001e-4

   1. Initial program 76.9%

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

      1. flip-+ 76.6%

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

      2. pow2 76.6%

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

      3. add-sqr-sqrt 77.7%

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

      4. *-commutative 77.7%

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

      5. *-commutative 77.7%

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

      6. *-commutative 77.7%

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

      7. *-commutative 77.7%

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

   3. Applied egg-rr 77.7%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 39.1%

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

      1. neg-sub0 39.1%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 39.1%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 39.1%

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

      4. neg-mul-1 39.1%

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

      5. associate-*l/ 39.1%

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

      6. *-commutative 39.1%

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

      7. associate-/r* 39.1%

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

      8. /-rgt-identity 39.1%

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

      9. metadata-eval 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in b around inf 91.4%

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

      1. distribute-lft-out 91.4%

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

      2. associate-/l* 91.5%

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

      3. associate-/l* 91.5%

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

      4. unpow2 91.5%

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

      5. unpow2 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in c around 0 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. associate-*r/ 91.8%

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

      5. neg-mul-1 91.8%

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

      6. associate-/l* 91.8%

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

      7. unpow2 91.8%

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

   9. Simplified 91.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.00025:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 9: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.00025:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.00025)
   (* (- b (sqrt (fma a (* c -4.0) (* b b)))) (/ -0.5 a))
   (- (/ (- c) b) (/ (* c c) (/ (pow b 3.0) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.00025) {
		tmp = (b - sqrt(fma(a, (c * -4.0), (b * b)))) * (-0.5 / a);
	} else {
		tmp = (-c / b) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.00025)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2.5000000000000001e-4

   1. Initial program 76.9%

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

      1. neg-sub0 76.9%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 76.9%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 76.9%

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

      4. neg-mul-1 76.9%

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

      5. associate-*l/ 76.9%

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

      6. *-commutative 76.9%

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

      7. associate-/r* 76.9%

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

      8. /-rgt-identity 76.9%

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

      9. metadata-eval 76.9%

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

   3. Simplified 77.0%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 39.1%

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

      1. neg-sub0 39.1%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 39.1%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 39.1%

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

      4. neg-mul-1 39.1%

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

      5. associate-*l/ 39.1%

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

      6. *-commutative 39.1%

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

      7. associate-/r* 39.1%

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

      8. /-rgt-identity 39.1%

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

      9. metadata-eval 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in b around inf 91.4%

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

      1. distribute-lft-out 91.4%

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

      2. associate-/l* 91.5%

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

      3. associate-/l* 91.5%

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

      4. unpow2 91.5%

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

      5. unpow2 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in c around 0 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. associate-*r/ 91.8%

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

      5. neg-mul-1 91.8%

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

      6. associate-/l* 91.8%

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

      7. unpow2 91.8%

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

   9. Simplified 91.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.00025:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 10: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.00025:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.00025)
   (* (- (sqrt (fma b b (* (* c a) -4.0))) b) (/ 0.5 a))
   (- (/ (- c) b) (/ (* c c) (/ (pow b 3.0) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.00025) {
		tmp = (sqrt(fma(b, b, ((c * a) * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = (-c / b) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.00025)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2.5000000000000001e-4

   1. Initial program 76.9%

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

      1. /-rgt-identity 76.9%

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

      2. metadata-eval 76.9%

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

      3. associate-/l* 76.9%

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

      4. associate-*r/ 76.9%

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

      5. +-commutative 76.9%

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

      6. unsub-neg 76.9%

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

      7. fma-neg 77.2%

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

      8. associate-*l* 77.2%

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

      9. *-commutative 77.2%

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

      10. distribute-rgt-neg-in 77.2%

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

      11. metadata-eval 77.2%

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

      12. associate-/r* 77.2%

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

      13. metadata-eval 77.2%

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

      14. metadata-eval 77.2%

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

   3. Simplified 77.2%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 39.1%

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

      1. neg-sub0 39.1%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 39.1%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 39.1%

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

      4. neg-mul-1 39.1%

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

      5. associate-*l/ 39.1%

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

      6. *-commutative 39.1%

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

      7. associate-/r* 39.1%

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

      8. /-rgt-identity 39.1%

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

      9. metadata-eval 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in b around inf 91.4%

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

      1. distribute-lft-out 91.4%

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

      2. associate-/l* 91.5%

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

      3. associate-/l* 91.5%

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

      4. unpow2 91.5%

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

      5. unpow2 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in c around 0 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. associate-*r/ 91.8%

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

      5. neg-mul-1 91.8%

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

      6. associate-/l* 91.8%

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

      7. unpow2 91.8%

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

   9. Simplified 91.8%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.00025:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 11: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.00025:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -0.00025)
   (* (/ 0.5 a) (- (sqrt (+ (* a (* c -4.0)) (* b b))) b))
   (- (/ (- c) b) (/ (* c c) (/ (pow b 3.0) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.00025) {
		tmp = (0.5 / a) * (sqrt(((a * (c * -4.0)) + (b * b))) - b);
	} else {
		tmp = (-c / b) - ((c * c) / (pow(b, 3.0) / a));
	}
	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 (((sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)) <= (-0.00025d0)) then
        tmp = (0.5d0 / a) * (sqrt(((a * (c * (-4.0d0))) + (b * b))) - b)
    else
        tmp = (-c / b) - ((c * c) / ((b ** 3.0d0) / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.00025) {
		tmp = (0.5 / a) * (Math.sqrt(((a * (c * -4.0)) + (b * b))) - b);
	} else {
		tmp = (-c / b) - ((c * c) / (Math.pow(b, 3.0) / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.00025:
		tmp = (0.5 / a) * (math.sqrt(((a * (c * -4.0)) + (b * b))) - b)
	else:
		tmp = (-c / b) - ((c * c) / (math.pow(b, 3.0) / a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -0.00025)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(a * Float64(c * -4.0)) + Float64(b * b))) - b));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -0.00025)
		tmp = (0.5 / a) * (sqrt(((a * (c * -4.0)) + (b * b))) - b);
	else
		tmp = (-c / b) - ((c * c) / ((b ^ 3.0) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.00025], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2.5000000000000001e-4

   1. Initial program 76.9%

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

      1. /-rgt-identity 76.9%

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

      2. metadata-eval 76.9%

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

      3. associate-/l* 76.9%

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

      4. associate-*r/ 76.9%

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

      5. +-commutative 76.9%

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

      6. unsub-neg 76.9%

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

      7. fma-neg 77.2%

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

      8. associate-*l* 77.2%

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

      9. *-commutative 77.2%

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

      10. distribute-rgt-neg-in 77.2%

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

      11. metadata-eval 77.2%

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

      12. associate-/r* 77.2%

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

      13. metadata-eval 77.2%

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

      14. metadata-eval 77.2%

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

   3. Simplified 77.2%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 76.9%

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

      2. associate-*l* 76.9%

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

   5. Applied egg-rr 76.9%

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

   if -2.5000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 39.1%

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

      1. neg-sub0 39.1%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

      2. associate-+l- 39.1%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

      3. sub0-neg 39.1%

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

      4. neg-mul-1 39.1%

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

      5. associate-*l/ 39.1%

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

      6. *-commutative 39.1%

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

      7. associate-/r* 39.1%

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

      8. /-rgt-identity 39.1%

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

      9. metadata-eval 39.1%

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

   3. Simplified 39.1%

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

   4. Taylor expanded in b around inf 91.4%

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

      1. distribute-lft-out 91.4%

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

      2. associate-/l* 91.5%

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

      3. associate-/l* 91.5%

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

      4. unpow2 91.5%

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

      5. unpow2 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in c around 0 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. associate-*r/ 91.8%

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

      5. neg-mul-1 91.8%

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

      6. associate-/l* 91.8%

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

      7. unpow2 91.8%

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

   9. Simplified 91.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -0.00025:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 12: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

   1. neg-sub0 55.4%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 55.4%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 55.4%

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

   4. neg-mul-1 55.4%

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

   5. associate-*l/ 55.4%

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

   6. *-commutative 55.4%

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

   7. associate-/r* 55.4%

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

   8. /-rgt-identity 55.4%

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

   9. metadata-eval 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in b around inf 80.6%

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

   1. distribute-lft-out 80.6%

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

   2. associate-/l* 80.6%

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

   3. associate-/l* 80.6%

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

   4. unpow2 80.6%

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

   5. unpow2 80.6%

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

6. Simplified 80.6%

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

7. Taylor expanded in c around 0 80.8%

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

   1. +-commutative 80.8%

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

   2. mul-1-neg 80.8%

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

   3. unsub-neg 80.8%

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

   4. associate-*r/ 80.8%

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

   5. neg-mul-1 80.8%

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

   6. associate-/l* 80.8%

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

   7. unpow2 80.8%

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

9. Simplified 80.8%

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

10. Final simplification 80.8%

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

Alternative 13: 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 55.4%

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

   1. neg-sub0 55.4%

      \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. associate-+l- 55.4%

      \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

   3. sub0-neg 55.4%

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

   4. neg-mul-1 55.4%

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

   5. associate-*l/ 55.4%

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

   6. *-commutative 55.4%

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

   7. associate-/r* 55.4%

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

   8. /-rgt-identity 55.4%

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

   9. metadata-eval 55.4%

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

3. Simplified 55.4%

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

4. Taylor expanded in b around inf 64.2%

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

   1. associate-*r/ 64.2%

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

   2. neg-mul-1 64.2%

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

6. Simplified 64.2%

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

7. Final simplification 64.2%

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

Alternative 14: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

   1. log1p-expm1-u 47.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)\right)} \]

   2. neg-mul-1 47.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)\right) \]

   3. fma-def 47.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\right)\right) \]

   4. *-commutative 47.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}{2 \cdot a}\right)\right) \]

   5. *-commutative 47.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}{2 \cdot a}\right)\right) \]

   6. *-commutative 47.8%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{\color{blue}{a \cdot 2}}\right)\right) \]

3. Applied egg-rr 47.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{a \cdot 2}\right)\right)} \]

4. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
5. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(b + -1 \cdot b\right)}{a}} \]

   2. distribute-rgt1-in 3.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

6. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

7. Final simplification 3.2%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023257 
(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)))