Quadratic roots, narrow range

Percentage Accurate: 55.6% → 92.0%

Time: 17.3s

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: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{a}^{3}}{{b}^{6}} \cdot -3.5\\ t_1 := \frac{{a}^{4}}{{b}^{8}}\\ t_2 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ t_3 := \frac{a \cdot a}{{b}^{4}} \cdot -1.5\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{t_2}^{1.5} - {b}^{3}}{t_2 + b \cdot \left(b + \sqrt{t_2}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left({c}^{3} \cdot \mathsf{fma}\left(-2, \frac{t_3}{b}, 2 \cdot \frac{t_0}{\frac{a}{b}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{t_3}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{t_1 \cdot 2.25}{\frac{a}{b}} + \mathsf{fma}\left(-2, \frac{t_0}{b}, 2 \cdot \frac{b}{\frac{a}{t_1 + t_1 \cdot -10.625}}\right)\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (pow a 3.0) (pow b 6.0)) -3.5))
        (t_1 (/ (pow a 4.0) (pow b 8.0)))
        (t_2 (fma b b (* c (* a -4.0))))
        (t_3 (* (/ (* a a) (pow b 4.0)) -1.5)))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -25.0)
     (/
      (/ (- (pow t_2 1.5) (pow b 3.0)) (+ t_2 (* b (+ b (sqrt t_2)))))
      (* a 2.0))
     (-
      (*
       0.5
       (+
        (+
         (* (pow c 3.0) (fma -2.0 (/ t_3 b) (* 2.0 (/ t_0 (/ a b)))))
         (* (* c c) (+ (/ a (pow b 3.0)) (* 2.0 (/ t_3 (/ a b))))))
        (*
         (pow c 4.0)
         (+
          (/ (* t_1 2.25) (/ a b))
          (fma -2.0 (/ t_0 b) (* 2.0 (/ b (/ a (+ t_1 (* t_1 -10.625))))))))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (pow(a, 3.0) / pow(b, 6.0)) * -3.5;
	double t_1 = pow(a, 4.0) / pow(b, 8.0);
	double t_2 = fma(b, b, (c * (a * -4.0)));
	double t_3 = ((a * a) / pow(b, 4.0)) * -1.5;
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -25.0) {
		tmp = ((pow(t_2, 1.5) - pow(b, 3.0)) / (t_2 + (b * (b + sqrt(t_2))))) / (a * 2.0);
	} else {
		tmp = (0.5 * (((pow(c, 3.0) * fma(-2.0, (t_3 / b), (2.0 * (t_0 / (a / b))))) + ((c * c) * ((a / pow(b, 3.0)) + (2.0 * (t_3 / (a / b)))))) + (pow(c, 4.0) * (((t_1 * 2.25) / (a / b)) + fma(-2.0, (t_0 / b), (2.0 * (b / (a / (t_1 + (t_1 * -10.625)))))))))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64((a ^ 3.0) / (b ^ 6.0)) * -3.5)
	t_1 = Float64((a ^ 4.0) / (b ^ 8.0))
	t_2 = fma(b, b, Float64(c * Float64(a * -4.0)))
	t_3 = Float64(Float64(Float64(a * a) / (b ^ 4.0)) * -1.5)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -25.0)
		tmp = Float64(Float64(Float64((t_2 ^ 1.5) - (b ^ 3.0)) / Float64(t_2 + Float64(b * Float64(b + sqrt(t_2))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64((c ^ 3.0) * fma(-2.0, Float64(t_3 / b), Float64(2.0 * Float64(t_0 / Float64(a / b))))) + Float64(Float64(c * c) * Float64(Float64(a / (b ^ 3.0)) + Float64(2.0 * Float64(t_3 / Float64(a / b)))))) + Float64((c ^ 4.0) * Float64(Float64(Float64(t_1 * 2.25) / Float64(a / b)) + fma(-2.0, Float64(t_0 / b), Float64(2.0 * Float64(b / Float64(a / Float64(t_1 + Float64(t_1 * -10.625)))))))))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * -3.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -1.5), $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], -25.0], N[(N[(N[(N[Power[t$95$2, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + N[(b * N[(b + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[(-2.0 * N[(t$95$3 / b), $MachinePrecision] + N[(2.0 * N[(t$95$0 / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$3 / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(N[(t$95$1 * 2.25), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(t$95$0 / b), $MachinePrecision] + N[(2.0 * N[(b / N[(a / N[(t$95$1 + N[(t$95$1 * -10.625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $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)) < -25

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. unsub-neg 87.7%

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

      4. fma-neg 87.8%

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

      5. associate-*l* 87.8%

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

      6. *-commutative 87.8%

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

      7. distribute-rgt-neg-in 87.8%

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

      8. metadata-eval 87.8%

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

   3. Simplified 87.8%

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

      1. fma-udef 87.7%

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

      2. *-commutative 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. flip3-- 88.1%

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

      2. fma-def 88.2%

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

      3. *-commutative 88.2%

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

      4. *-commutative 88.2%

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

      5. add-sqr-sqrt 88.2%

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

      6. fma-def 88.2%

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

      7. *-commutative 88.2%

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

      8. *-commutative 88.2%

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

   7. Applied egg-rr 88.3%

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

      1. cube-mult 88.5%

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

      2. unpow1/2 88.5%

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

      3. rem-square-sqrt 88.9%

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

      4. pow-plus 89.2%

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

      5. associate-*l* 89.2%

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

      6. metadata-eval 89.2%

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

      7. associate-*l* 89.2%

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

      8. distribute-rgt-out 89.3%

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

      9. associate-*l* 89.3%

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

   9. Simplified 89.3%

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

   if -25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

      2. add-sqr-sqrt 50.5%

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

      3. fma-def 51.0%

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

      4. *-commutative 51.0%

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

      5. *-commutative 51.0%

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

      6. *-commutative 51.0%

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

      7. *-commutative 51.0%

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

   3. Applied egg-rr 51.0%

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

   4. Taylor expanded in c around 0 94.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(0.5 \cdot \left({c}^{4} \cdot \left(\frac{{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right)}^{2} \cdot b}{a} + \left(-2 \cdot \frac{-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)}{b} + 2 \cdot \frac{b \cdot \left(2 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-1 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(5.333333333333333 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-16 \cdot \frac{{a}^{4}}{{b}^{8}} + 0.041666666666666664 \cdot \frac{{a}^{4}}{{b}^{8}}\right)\right)\right)\right)}{a}\right)\right)\right) + \left(0.5 \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}}{b} + 2 \cdot \frac{\left(-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)\right) \cdot b}{a}\right)\right) + 0.5 \cdot \left({c}^{2} \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) \cdot b}{a}\right)\right)\right)\right)} \]

   6. Simplified 94.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left({c}^{3} \cdot \mathsf{fma}\left(-2, \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{b}, 2 \cdot \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{\frac{a}{b}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + \mathsf{fma}\left(-2, \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{b}, 2 \cdot \frac{b}{\frac{a}{\frac{{a}^{4}}{{b}^{8}} \cdot 1 + \frac{{a}^{4}}{{b}^{8}} \cdot -10.625}}\right)\right)\right) - \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left({c}^{3} \cdot \mathsf{fma}\left(-2, \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{b}, 2 \cdot \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{\frac{a}{b}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + \mathsf{fma}\left(-2, \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{b}, 2 \cdot \frac{b}{\frac{a}{\frac{{a}^{4}}{{b}^{8}} + \frac{{a}^{4}}{{b}^{8}} \cdot -10.625}}\right)\right)\right) - \frac{c}{b}\\ \end{array} \]

Alternative 2: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{t_0}^{1.5} - {b}^{3}}{t_0 + b \cdot \left(b + \sqrt{t_0}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right) + {c}^{3} \cdot \frac{-4 \cdot \left(a \cdot a\right)}{{b}^{5}}\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -25.0)
     (/
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ t_0 (* b (+ b (sqrt t_0)))))
      (* a 2.0))
     (-
      (*
       0.5
       (+
        (+
         (*
          (* c c)
          (+
           (/ a (pow b 3.0))
           (* 2.0 (/ (* (/ (* a a) (pow b 4.0)) -1.5) (/ a b)))))
         (* (pow c 3.0) (/ (* -4.0 (* a a)) (pow b 5.0))))
        (*
         (pow c 4.0)
         (+
          (/ (* (/ (pow a 4.0) (pow b 8.0)) 2.25) (/ a b))
          (* -12.25 (/ (pow a 3.0) (pow b 7.0)))))))
      (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -25.0) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) / (t_0 + (b * (b + sqrt(t_0))))) / (a * 2.0);
	} else {
		tmp = (0.5 * ((((c * c) * ((a / pow(b, 3.0)) + (2.0 * ((((a * a) / pow(b, 4.0)) * -1.5) / (a / b))))) + (pow(c, 3.0) * ((-4.0 * (a * a)) / pow(b, 5.0)))) + (pow(c, 4.0) * ((((pow(a, 4.0) / pow(b, 8.0)) * 2.25) / (a / b)) + (-12.25 * (pow(a, 3.0) / pow(b, 7.0))))))) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -25.0)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + Float64(b * Float64(b + sqrt(t_0))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(Float64(Float64(c * c) * Float64(Float64(a / (b ^ 3.0)) + Float64(2.0 * Float64(Float64(Float64(Float64(a * a) / (b ^ 4.0)) * -1.5) / Float64(a / b))))) + Float64((c ^ 3.0) * Float64(Float64(-4.0 * Float64(a * a)) / (b ^ 5.0)))) + Float64((c ^ 4.0) * Float64(Float64(Float64(Float64((a ^ 4.0) / (b ^ 8.0)) * 2.25) / Float64(a / b)) + Float64(-12.25 * Float64((a ^ 3.0) / (b ^ 7.0))))))) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -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], -25.0], 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[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(N[(N[(c * c), $MachinePrecision] * N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision] * 2.25), $MachinePrecision] / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(-12.25 * N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $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)) < -25

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. unsub-neg 87.7%

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

      4. fma-neg 87.8%

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

      5. associate-*l* 87.8%

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

      6. *-commutative 87.8%

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

      7. distribute-rgt-neg-in 87.8%

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

      8. metadata-eval 87.8%

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

   3. Simplified 87.8%

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

      1. fma-udef 87.7%

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

      2. *-commutative 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. flip3-- 88.1%

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

      2. fma-def 88.2%

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

      3. *-commutative 88.2%

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

      4. *-commutative 88.2%

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

      5. add-sqr-sqrt 88.2%

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

      6. fma-def 88.2%

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

      7. *-commutative 88.2%

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

      8. *-commutative 88.2%

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

   7. Applied egg-rr 88.3%

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

      1. cube-mult 88.5%

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

      2. unpow1/2 88.5%

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

      3. rem-square-sqrt 88.9%

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

      4. pow-plus 89.2%

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

      5. associate-*l* 89.2%

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

      6. metadata-eval 89.2%

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

      7. associate-*l* 89.2%

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

      8. distribute-rgt-out 89.3%

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

      9. associate-*l* 89.3%

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

   9. Simplified 89.3%

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

   if -25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.3%

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

      1. +-commutative 51.3%

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

      2. add-sqr-sqrt 50.5%

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

      3. fma-def 51.0%

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

      4. *-commutative 51.0%

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

      5. *-commutative 51.0%

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

      6. *-commutative 51.0%

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

      7. *-commutative 51.0%

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

   3. Applied egg-rr 51.0%

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

   4. Taylor expanded in c around 0 94.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + \left(0.5 \cdot \left({c}^{4} \cdot \left(\frac{{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right)}^{2} \cdot b}{a} + \left(-2 \cdot \frac{-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)}{b} + 2 \cdot \frac{b \cdot \left(2 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-1 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(5.333333333333333 \cdot \frac{{a}^{4}}{{b}^{8}} + \left(-16 \cdot \frac{{a}^{4}}{{b}^{8}} + 0.041666666666666664 \cdot \frac{{a}^{4}}{{b}^{8}}\right)\right)\right)\right)}{a}\right)\right)\right) + \left(0.5 \cdot \left({c}^{3} \cdot \left(-2 \cdot \frac{0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}}{b} + 2 \cdot \frac{\left(-0.16666666666666666 \cdot \frac{{a}^{3}}{{b}^{6}} + \left(2 \cdot \frac{{a}^{3}}{{b}^{6}} + -5.333333333333333 \cdot \frac{{a}^{3}}{{b}^{6}}\right)\right) \cdot b}{a}\right)\right) + 0.5 \cdot \left({c}^{2} \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\left(0.5 \cdot \frac{{a}^{2}}{{b}^{4}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) \cdot b}{a}\right)\right)\right)\right)} \]

   6. Simplified 94.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left({c}^{3} \cdot \mathsf{fma}\left(-2, \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{b}, 2 \cdot \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{\frac{a}{b}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + \mathsf{fma}\left(-2, \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{b}, 2 \cdot \frac{b}{\frac{a}{\frac{{a}^{4}}{{b}^{8}} \cdot 1 + \frac{{a}^{4}}{{b}^{8}} \cdot -10.625}}\right)\right)\right) - \frac{c}{b}} \]

   7. Taylor expanded in a around 0 94.2%

      \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \mathsf{fma}\left(-2, \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{b}, 2 \cdot \frac{\frac{{a}^{3}}{{b}^{6}} \cdot -3.5}{\frac{a}{b}}\right) + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + \color{blue}{-12.25 \cdot \frac{{a}^{3}}{{b}^{7}}}\right)\right) - \frac{c}{b} \]

   8. Taylor expanded in b around 0 94.2%

      \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \color{blue}{\frac{3 \cdot {a}^{2} + -7 \cdot {a}^{2}}{{b}^{5}}} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]
   9. Step-by-step derivation

      1. distribute-rgt-out 94.2%

         \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \frac{\color{blue}{{a}^{2} \cdot \left(3 + -7\right)}}{{b}^{5}} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]

      2. unpow2 94.2%

         \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \frac{\color{blue}{\left(a \cdot a\right)} \cdot \left(3 + -7\right)}{{b}^{5}} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]

      3. metadata-eval 94.2%

         \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \frac{\left(a \cdot a\right) \cdot \color{blue}{-4}}{{b}^{5}} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]

   10. Simplified 94.2%

       \[\leadsto 0.5 \cdot \left(\left({c}^{3} \cdot \color{blue}{\frac{\left(a \cdot a\right) \cdot -4}{{b}^{5}}} + \left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right)\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\frac{a}{{b}^{3}} + 2 \cdot \frac{\frac{a \cdot a}{{b}^{4}} \cdot -1.5}{\frac{a}{b}}\right) + {c}^{3} \cdot \frac{-4 \cdot \left(a \cdot a\right)}{{b}^{5}}\right) + {c}^{4} \cdot \left(\frac{\frac{{a}^{4}}{{b}^{8}} \cdot 2.25}{\frac{a}{b}} + -12.25 \cdot \frac{{a}^{3}}{{b}^{7}}\right)\right) - \frac{c}{b}\\ \end{array} \]

Alternative 3: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{t_0}^{1.5} - {b}^{3}}{t_0 + b \cdot \left(b + \sqrt{t_0}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{20}{\frac{{b}^{7}}{{a}^{3} \cdot {c}^{4}}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -25.0)
     (/
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ t_0 (* b (+ b (sqrt t_0)))))
      (* a 2.0))
     (-
      (-
       (fma
        -0.25
        (/ 20.0 (/ (pow b 7.0) (* (pow a 3.0) (pow c 4.0))))
        (* -2.0 (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))))
       (/ c b))
      (/ (* c c) (/ (pow b 3.0) a))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -25.0) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) / (t_0 + (b * (b + sqrt(t_0))))) / (a * 2.0);
	} else {
		tmp = (fma(-0.25, (20.0 / (pow(b, 7.0) / (pow(a, 3.0) * pow(c, 4.0)))), (-2.0 * (pow(c, 3.0) / (pow(b, 5.0) / (a * a))))) - (c / b)) - ((c * c) / (pow(b, 3.0) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -25.0)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + Float64(b * Float64(b + sqrt(t_0))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(-0.25, Float64(20.0 / Float64((b ^ 7.0) / Float64((a ^ 3.0) * (c ^ 4.0)))), Float64(-2.0 * Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))))) - Float64(c / b)) - Float64(Float64(c * c) / Float64((b ^ 3.0) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -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], -25.0], 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[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * N[(20.0 / N[(N[Power[b, 7.0], $MachinePrecision] / N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $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)) < -25

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. unsub-neg 87.7%

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

      4. fma-neg 87.8%

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

      5. associate-*l* 87.8%

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

      6. *-commutative 87.8%

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

      7. distribute-rgt-neg-in 87.8%

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

      8. metadata-eval 87.8%

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

   3. Simplified 87.8%

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

      1. fma-udef 87.7%

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

      2. *-commutative 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. flip3-- 88.1%

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

      2. fma-def 88.2%

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

      3. *-commutative 88.2%

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

      4. *-commutative 88.2%

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

      5. add-sqr-sqrt 88.2%

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

      6. fma-def 88.2%

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

      7. *-commutative 88.2%

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

      8. *-commutative 88.2%

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

   7. Applied egg-rr 88.3%

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

      1. cube-mult 88.5%

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

      2. unpow1/2 88.5%

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

      3. rem-square-sqrt 88.9%

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

      4. pow-plus 89.2%

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

      5. associate-*l* 89.2%

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

      6. metadata-eval 89.2%

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

      7. associate-*l* 89.2%

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

      8. distribute-rgt-out 89.3%

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

      9. associate-*l* 89.3%

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

   9. Simplified 89.3%

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

   if -25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.3%

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

      1. *-commutative 51.3%

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

      2. +-commutative 51.3%

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

      3. unsub-neg 51.3%

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

      4. fma-neg 51.5%

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

      5. associate-*l* 51.5%

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

      6. *-commutative 51.5%

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

      7. distribute-rgt-neg-in 51.5%

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

      8. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

      1. fma-udef 51.3%

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

      2. *-commutative 51.3%

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

   5. Applied egg-rr 51.3%

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

      1. add-cbrt-cube 51.3%

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

      2. fma-def 51.6%

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

      3. *-commutative 51.6%

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

      4. *-commutative 51.6%

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

      5. fma-def 51.6%

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

      6. *-commutative 51.6%

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

      7. *-commutative 51.6%

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

   7. Applied egg-rr 51.5%

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

      1. associate-*l* 51.5%

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

      2. metadata-eval 51.5%

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

      3. distribute-rgt-neg-in 51.5%

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

      4. associate-*r* 51.5%

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

      5. fma-neg 51.6%

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

      6. unsub-neg 51.6%

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

   9. Simplified 51.5%

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

   10. Taylor expanded in a around 0 94.2%

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

   12. Simplified 94.2%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.25, \frac{{a}^{3}}{\frac{b}{\frac{{c}^{4}}{{b}^{6}} \cdot 20}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]

   13. Taylor expanded in a around 0 94.2%

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

       1. associate-*r/ 94.2%

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

       2. associate-/l* 94.2%

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

       3. *-commutative 94.2%

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

   15. Simplified 94.2%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.25, \frac{20}{\frac{{b}^{7}}{{a}^{3} \cdot {c}^{4}}}, -2 \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}\right) - \frac{c}{b}\right) - \frac{c \cdot c}{\frac{{b}^{3}}{a}}\\ \end{array} \]

Alternative 4: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{{t_0}^{1.5} - {b}^{3}}{\left(a \cdot 2\right) \cdot \left(t_0 + b \cdot \left(b + \sqrt{t_0}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* -4.0 (* a c)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -25.0)
     (/
      (- (pow t_0 1.5) (pow b 3.0))
      (* (* a 2.0) (+ t_0 (* b (+ b (sqrt t_0))))))
     (-
      (- (/ (* -2.0 (* (pow c 3.0) (* a a))) (pow b 5.0)) (/ c b))
      (/ (* c (* a c)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (-4.0 * (a * c)));
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -25.0) {
		tmp = (pow(t_0, 1.5) - pow(b, 3.0)) / ((a * 2.0) * (t_0 + (b * (b + sqrt(t_0)))));
	} else {
		tmp = (((-2.0 * (pow(c, 3.0) * (a * a))) / pow(b, 5.0)) - (c / b)) - ((c * (a * c)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(-4.0 * Float64(a * c)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -25.0)
		tmp = Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(Float64(a * 2.0) * Float64(t_0 + Float64(b * Float64(b + sqrt(t_0))))));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((c ^ 3.0) * Float64(a * a))) / (b ^ 5.0)) - Float64(c / b)) - Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(-4.0 * N[(a * c), $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], -25.0], N[(N[(N[Power[t$95$0, 1.5], $MachinePrecision] - N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[(t$95$0 + N[(b * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(a * c), $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)) < -25

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. unsub-neg 87.7%

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

      4. fma-neg 87.8%

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

      5. associate-*l* 87.8%

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

      6. *-commutative 87.8%

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

      7. distribute-rgt-neg-in 87.8%

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

      8. metadata-eval 87.8%

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

   3. Simplified 87.8%

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

      1. fma-udef 87.7%

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

      2. *-commutative 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. flip3-- 88.1%

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

      2. fma-def 88.2%

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

      3. *-commutative 88.2%

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

      4. *-commutative 88.2%

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

      5. add-sqr-sqrt 88.2%

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

      6. fma-def 88.2%

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

      7. *-commutative 88.2%

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

      8. *-commutative 88.2%

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

   7. Applied egg-rr 88.3%

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

      1. cube-mult 88.5%

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

      2. unpow1/2 88.5%

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

      3. rem-square-sqrt 88.9%

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

      4. pow-plus 89.2%

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

      5. associate-*l* 89.2%

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

      6. metadata-eval 89.2%

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

      7. associate-*l* 89.2%

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

      8. distribute-rgt-out 89.3%

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

      9. associate-*l* 89.3%

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

   9. Simplified 89.3%

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

       1. *-un-lft-identity 89.3%

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

   11. Applied egg-rr 89.3%

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

       1. *-lft-identity 89.3%

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

       2. associate-/l/ 89.3%

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

       3. associate-*r* 89.3%

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

       4. *-commutative 89.3%

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

       5. *-commutative 89.3%

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

       6. associate-*r* 89.3%

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

       7. *-commutative 89.3%

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

       8. *-commutative 89.3%

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

       9. associate-*r* 89.3%

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

       10. *-commutative 89.3%

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

       11. *-commutative 89.3%

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

   13. Simplified 89.3%

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

   if -25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.3%

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

      1. neg-sub0 51.3%

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

      2. associate-+l- 51.3%

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

      3. sub0-neg 51.3%

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

      4. neg-mul-1 51.3%

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

      5. associate-*l/ 51.3%

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

      6. *-commutative 51.3%

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

      7. associate-/r* 51.3%

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

      8. /-rgt-identity 51.3%

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

      9. metadata-eval 51.3%

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

   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 91.3%

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

      1. +-commutative 91.3%

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

      2. mul-1-neg 91.3%

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

      3. unsub-neg 91.3%

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

      4. +-commutative 91.3%

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

      5. mul-1-neg 91.3%

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

      6. unsub-neg 91.3%

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

      7. associate-*r/ 91.3%

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

      8. unpow2 91.3%

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

      9. unpow2 91.3%

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

      10. associate-*l* 91.3%

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

   6. Simplified 91.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)\right)}^{1.5} - {b}^{3}}{\left(a \cdot 2\right) \cdot \left(\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 5: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{t_0}^{1.5} - {b}^{3}}{t_0 + b \cdot \left(b + \sqrt{t_0}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* c (* a -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0)) -25.0)
     (/
      (/ (- (pow t_0 1.5) (pow b 3.0)) (+ t_0 (* b (+ b (sqrt t_0)))))
      (* a 2.0))
     (-
      (- (/ (* -2.0 (* (pow c 3.0) (* a a))) (pow b 5.0)) (/ c b))
      (/ (* c (* a c)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (c * (a * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)) <= -25.0) {
		tmp = ((pow(t_0, 1.5) - pow(b, 3.0)) / (t_0 + (b * (b + sqrt(t_0))))) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (pow(c, 3.0) * (a * a))) / pow(b, 5.0)) - (c / b)) - ((c * (a * c)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(c * Float64(a * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0)) <= -25.0)
		tmp = Float64(Float64(Float64((t_0 ^ 1.5) - (b ^ 3.0)) / Float64(t_0 + Float64(b * Float64(b + sqrt(t_0))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((c ^ 3.0) * Float64(a * a))) / (b ^ 5.0)) - Float64(c / b)) - Float64(Float64(c * Float64(a * c)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(c * N[(a * -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], -25.0], 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[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(a * c), $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)) < -25

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. +-commutative 87.7%

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

      3. unsub-neg 87.7%

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

      4. fma-neg 87.8%

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

      5. associate-*l* 87.8%

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

      6. *-commutative 87.8%

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

      7. distribute-rgt-neg-in 87.8%

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

      8. metadata-eval 87.8%

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

   3. Simplified 87.8%

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

      1. fma-udef 87.7%

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

      2. *-commutative 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. flip3-- 88.1%

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

      2. fma-def 88.2%

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

      3. *-commutative 88.2%

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

      4. *-commutative 88.2%

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

      5. add-sqr-sqrt 88.2%

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

      6. fma-def 88.2%

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

      7. *-commutative 88.2%

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

      8. *-commutative 88.2%

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

   7. Applied egg-rr 88.3%

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

      1. cube-mult 88.5%

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

      2. unpow1/2 88.5%

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

      3. rem-square-sqrt 88.9%

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

      4. pow-plus 89.2%

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

      5. associate-*l* 89.2%

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

      6. metadata-eval 89.2%

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

      7. associate-*l* 89.2%

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

      8. distribute-rgt-out 89.3%

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

      9. associate-*l* 89.3%

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

   9. Simplified 89.3%

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

   if -25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.3%

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

      1. neg-sub0 51.3%

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

      2. associate-+l- 51.3%

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

      3. sub0-neg 51.3%

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

      4. neg-mul-1 51.3%

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

      5. associate-*l/ 51.3%

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

      6. *-commutative 51.3%

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

      7. associate-/r* 51.3%

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

      8. /-rgt-identity 51.3%

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

      9. metadata-eval 51.3%

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

   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 91.3%

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

      1. +-commutative 91.3%

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

      2. mul-1-neg 91.3%

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

      3. unsub-neg 91.3%

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

      4. +-commutative 91.3%

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

      5. mul-1-neg 91.3%

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

      6. unsub-neg 91.3%

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

      7. associate-*r/ 91.3%

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

      8. unpow2 91.3%

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

      9. unpow2 91.3%

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

      10. associate-*l* 91.3%

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

   6. Simplified 91.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5} - {b}^{3}}{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right) + b \cdot \left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 6: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t_0}^{1.5}}{{\left(-b\right)}^{2} + \left(t_0 + b \cdot t_1\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

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)) -25.0)
     (/
      (/
       (+ (pow (- b) 3.0) (pow t_0 1.5))
       (+ (pow (- b) 2.0) (+ t_0 (* b t_1))))
      (* a 2.0))
     (-
      (- (/ (* -2.0 (* (pow c 3.0) (* a a))) (pow b 5.0)) (/ c b))
      (/ (* c (* a c)) (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)) <= -25.0) {
		tmp = ((pow(-b, 3.0) + pow(t_0, 1.5)) / (pow(-b, 2.0) + (t_0 + (b * t_1)))) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (pow(c, 3.0) * (a * a))) / pow(b, 5.0)) - (c / b)) - ((c * (a * c)) / 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)) <= (-25.0d0)) then
        tmp = (((-b ** 3.0d0) + (t_0 ** 1.5d0)) / ((-b ** 2.0d0) + (t_0 + (b * t_1)))) / (a * 2.0d0)
    else
        tmp = ((((-2.0d0) * ((c ** 3.0d0) * (a * a))) / (b ** 5.0d0)) - (c / b)) - ((c * (a * c)) / (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)) <= -25.0) {
		tmp = ((Math.pow(-b, 3.0) + Math.pow(t_0, 1.5)) / (Math.pow(-b, 2.0) + (t_0 + (b * t_1)))) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (Math.pow(c, 3.0) * (a * a))) / Math.pow(b, 5.0)) - (c / b)) - ((c * (a * c)) / 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)) <= -25.0:
		tmp = ((math.pow(-b, 3.0) + math.pow(t_0, 1.5)) / (math.pow(-b, 2.0) + (t_0 + (b * t_1)))) / (a * 2.0)
	else:
		tmp = (((-2.0 * (math.pow(c, 3.0) * (a * a))) / math.pow(b, 5.0)) - (c / b)) - ((c * (a * c)) / 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)) <= -25.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_0 ^ 1.5)) / Float64((Float64(-b) ^ 2.0) + Float64(t_0 + Float64(b * t_1)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((c ^ 3.0) * Float64(a * a))) / (b ^ 5.0)) - Float64(c / b)) - Float64(Float64(c * Float64(a * c)) / (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)) <= -25.0)
		tmp = (((-b ^ 3.0) + (t_0 ^ 1.5)) / ((-b ^ 2.0) + (t_0 + (b * t_1)))) / (a * 2.0);
	else
		tmp = (((-2.0 * ((c ^ 3.0) * (a * a))) / (b ^ 5.0)) - (c / b)) - ((c * (a * c)) / (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], -25.0], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Power[(-b), 2.0], $MachinePrecision] + N[(t$95$0 + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(a * c), $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)) < -25

   1. Initial program 87.7%

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

      1. flip3-+ 88.1%

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

      2. pow1/2 88.1%

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

      3. pow-pow 88.8%

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

      4. *-commutative 88.8%

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

      5. *-commutative 88.8%

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

      6. metadata-eval 88.8%

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

      7. pow2 88.8%

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

   3. Applied egg-rr 88.8%

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

   if -25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.3%

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

      1. neg-sub0 51.3%

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

      2. associate-+l- 51.3%

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

      3. sub0-neg 51.3%

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

      4. neg-mul-1 51.3%

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

      5. associate-*l/ 51.3%

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

      6. *-commutative 51.3%

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

      7. associate-/r* 51.3%

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

      8. /-rgt-identity 51.3%

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

      9. metadata-eval 51.3%

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

   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 91.3%

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

      1. +-commutative 91.3%

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

      2. mul-1-neg 91.3%

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

      3. unsub-neg 91.3%

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

      4. +-commutative 91.3%

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

      5. mul-1-neg 91.3%

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

      6. unsub-neg 91.3%

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

      7. associate-*r/ 91.3%

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

      8. unpow2 91.3%

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

      9. unpow2 91.3%

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

      10. associate-*l* 91.3%

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

   6. Simplified 91.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}^{1.5}}{{\left(-b\right)}^{2} + \left(\left(b \cdot b - c \cdot \left(4 \cdot a\right)\right) + b \cdot \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 7: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(4 \cdot a\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;\frac{t_1 - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - t_0}{\left(-b\right) - t_1}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

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)) -25.0)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) t_1)) (* a 2.0))
     (-
      (- (/ (* -2.0 (* (pow c 3.0) (* a a))) (pow b 5.0)) (/ c b))
      (/ (* c (* a c)) (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)) <= -25.0) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (pow(c, 3.0) * (a * a))) / pow(b, 5.0)) - (c / b)) - ((c * (a * c)) / 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)) <= (-25.0d0)) then
        tmp = (((-b ** 2.0d0) - t_0) / (-b - t_1)) / (a * 2.0d0)
    else
        tmp = ((((-2.0d0) * ((c ** 3.0d0) * (a * a))) / (b ** 5.0d0)) - (c / b)) - ((c * (a * c)) / (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)) <= -25.0) {
		tmp = ((Math.pow(-b, 2.0) - t_0) / (-b - t_1)) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (Math.pow(c, 3.0) * (a * a))) / Math.pow(b, 5.0)) - (c / b)) - ((c * (a * c)) / 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)) <= -25.0:
		tmp = ((math.pow(-b, 2.0) - t_0) / (-b - t_1)) / (a * 2.0)
	else:
		tmp = (((-2.0 * (math.pow(c, 3.0) * (a * a))) / math.pow(b, 5.0)) - (c / b)) - ((c * (a * c)) / 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)) <= -25.0)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - t_1)) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((c ^ 3.0) * Float64(a * a))) / (b ^ 5.0)) - Float64(c / b)) - Float64(Float64(c * Float64(a * c)) / (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)) <= -25.0)
		tmp = (((-b ^ 2.0) - t_0) / (-b - t_1)) / (a * 2.0);
	else
		tmp = (((-2.0 * ((c ^ 3.0) * (a * a))) / (b ^ 5.0)) - (c / b)) - ((c * (a * c)) / (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], -25.0], 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[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * N[(a * c), $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)) < -25

   1. Initial program 87.7%

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

      1. flip-+ 87.4%

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

      2. pow2 87.4%

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

      3. add-sqr-sqrt 88.8%

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

      4. *-commutative 88.8%

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

      5. *-commutative 88.8%

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

      6. *-commutative 88.8%

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

      7. *-commutative 88.8%

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

   3. Applied egg-rr 88.8%

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

   if -25 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 51.3%

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

      1. neg-sub0 51.3%

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

      2. associate-+l- 51.3%

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

      3. sub0-neg 51.3%

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

      4. neg-mul-1 51.3%

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

      5. associate-*l/ 51.3%

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

      6. *-commutative 51.3%

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

      7. associate-/r* 51.3%

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

      8. /-rgt-identity 51.3%

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

      9. metadata-eval 51.3%

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

   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 91.3%

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

      1. +-commutative 91.3%

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

      2. mul-1-neg 91.3%

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

      3. unsub-neg 91.3%

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

      4. +-commutative 91.3%

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

      5. mul-1-neg 91.3%

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

      6. unsub-neg 91.3%

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

      7. associate-*r/ 91.3%

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

      8. unpow2 91.3%

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

      9. unpow2 91.3%

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

      10. associate-*l* 91.3%

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

   6. Simplified 91.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -25:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({c}^{3} \cdot \left(a \cdot a\right)\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 8: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - c \cdot \left(4 \cdot a\right)\\ \mathbf{if}\;b \leq 2.75:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - t_0}{\left(-b\right) - \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* c (* 4.0 a)))))
   (if (<= b 2.75)
     (/ (/ (- (pow (- b) 2.0) t_0) (- (- b) (sqrt t_0))) (* a 2.0))
     (- (/ (- c) b) (/ (* a (* c c)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - (c * (4.0 * a));
	double tmp;
	if (b <= 2.75) {
		tmp = ((pow(-b, 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * (c * c)) / pow(b, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (c * (4.0d0 * a))
    if (b <= 2.75d0) then
        tmp = (((-b ** 2.0d0) - t_0) / (-b - sqrt(t_0))) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a * (c * c)) / (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 tmp;
	if (b <= 2.75) {
		tmp = ((Math.pow(-b, 2.0) - t_0) / (-b - Math.sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * (c * c)) / Math.pow(b, 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) - (c * (4.0 * a))
	tmp = 0
	if b <= 2.75:
		tmp = ((math.pow(-b, 2.0) - t_0) / (-b - math.sqrt(t_0))) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a * (c * c)) / 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)))
	tmp = 0.0
	if (b <= 2.75)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 2.0) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) - (c * (4.0 * a));
	tmp = 0.0;
	if (b <= 2.75)
		tmp = (((-b ^ 2.0) - t_0) / (-b - sqrt(t_0))) / (a * 2.0);
	else
		tmp = (-c / b) - ((a * (c * c)) / (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]}, If[LessEqual[b, 2.75], N[(N[(N[(N[Power[(-b), 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.75

   1. Initial program 82.2%

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

      1. flip-+ 82.0%

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

      2. pow2 82.0%

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

      3. add-sqr-sqrt 83.6%

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

      4. *-commutative 83.6%

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

      5. *-commutative 83.6%

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

      6. *-commutative 83.6%

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

      7. *-commutative 83.6%

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

   3. Applied egg-rr 83.6%

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

   if 2.75 < b

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

      2. +-commutative 47.5%

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

      3. unsub-neg 47.5%

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

      4. fma-neg 47.7%

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

      5. associate-*l* 47.7%

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

      6. *-commutative 47.7%

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

      7. distribute-rgt-neg-in 47.7%

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

      8. metadata-eval 47.7%

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

   3. Simplified 47.7%

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

      1. fma-udef 47.5%

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

      2. *-commutative 47.5%

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

   5. Applied egg-rr 47.5%

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

   6. Taylor expanded in b around inf 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unpow2 87.9%

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

      4. associate-*r* 87.9%

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

      5. unsub-neg 87.9%

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

      6. associate-*r/ 87.9%

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

      7. neg-mul-1 87.9%

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

      8. associate-*r* 87.9%

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

      9. unpow2 87.9%

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

      10. *-commutative 87.9%

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

      11. unpow2 87.9%

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

   8. Simplified 87.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.75:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{2} - \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 9: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.5)
   (* (- b (sqrt (fma a (* c -4.0) (* b b)))) (/ -0.5 a))
   (- (/ (- c) b) (/ (* a (* c c)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5) {
		tmp = (b - sqrt(fma(a, (c * -4.0), (b * b)))) * (-0.5 / a);
	} else {
		tmp = (-c / b) - ((a * (c * c)) / pow(b, 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.5)
		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(a * Float64(c * c)) / (b ^ 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.5], 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[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5

   1. Initial program 82.2%

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

      1. neg-sub0 82.2%

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

      2. associate-+l- 82.2%

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

      3. sub0-neg 82.2%

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

      4. neg-mul-1 82.2%

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

      5. associate-*l/ 82.2%

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

      6. *-commutative 82.2%

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

      7. associate-/r* 82.2%

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

      8. /-rgt-identity 82.2%

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

      9. metadata-eval 82.2%

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

   3. Simplified 82.3%

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

   if 2.5 < b

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

      2. +-commutative 47.5%

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

      3. unsub-neg 47.5%

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

      4. fma-neg 47.7%

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

      5. associate-*l* 47.7%

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

      6. *-commutative 47.7%

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

      7. distribute-rgt-neg-in 47.7%

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

      8. metadata-eval 47.7%

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

   3. Simplified 47.7%

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

      1. fma-udef 47.5%

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

      2. *-commutative 47.5%

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

   5. Applied egg-rr 47.5%

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

   6. Taylor expanded in b around inf 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unpow2 87.9%

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

      4. associate-*r* 87.9%

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

      5. unsub-neg 87.9%

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

      6. associate-*r/ 87.9%

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

      7. neg-mul-1 87.9%

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

      8. associate-*r* 87.9%

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

      9. unpow2 87.9%

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

      10. *-commutative 87.9%

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

      11. unpow2 87.9%

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

   8. Simplified 87.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 10: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.55:\\ \;\;\;\;\left(\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.55)
   (* (- (sqrt (+ (* b b) (* -4.0 (* a c)))) b) (/ 0.5 a))
   (- (/ (- c) b) (/ (* a (* c c)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.55) {
		tmp = (sqrt(((b * b) + (-4.0 * (a * c)))) - b) * (0.5 / a);
	} else {
		tmp = (-c / b) - ((a * (c * c)) / pow(b, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.55d0) then
        tmp = (sqrt(((b * b) + ((-4.0d0) * (a * c)))) - b) * (0.5d0 / a)
    else
        tmp = (-c / b) - ((a * (c * c)) / (b ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.55) {
		tmp = (Math.sqrt(((b * b) + (-4.0 * (a * c)))) - b) * (0.5 / a);
	} else {
		tmp = (-c / b) - ((a * (c * c)) / Math.pow(b, 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.55:
		tmp = (math.sqrt(((b * b) + (-4.0 * (a * c)))) - b) * (0.5 / a)
	else:
		tmp = (-c / b) - ((a * (c * c)) / math.pow(b, 3.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.55)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(a * c)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.55)
		tmp = (sqrt(((b * b) + (-4.0 * (a * c)))) - b) * (0.5 / a);
	else
		tmp = (-c / b) - ((a * (c * c)) / (b ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.55], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5499999999999998

   1. Initial program 82.2%

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

      1. /-rgt-identity 82.2%

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

      2. metadata-eval 82.2%

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

      3. associate-/l* 82.2%

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

      4. associate-*r/ 82.2%

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

      5. +-commutative 82.2%

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

      6. unsub-neg 82.2%

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

      7. fma-neg 82.2%

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

      8. associate-*l* 82.2%

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

      9. *-commutative 82.2%

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

      10. distribute-rgt-neg-in 82.2%

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

      11. metadata-eval 82.2%

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

      12. associate-/r* 82.2%

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

      13. metadata-eval 82.2%

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

      14. metadata-eval 82.2%

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

   3. Simplified 82.2%

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

      1. fma-udef 82.2%

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

      2. *-commutative 82.2%

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

   5. Applied egg-rr 82.2%

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

   if 2.5499999999999998 < b

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

      2. +-commutative 47.5%

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

      3. unsub-neg 47.5%

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

      4. fma-neg 47.7%

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

      5. associate-*l* 47.7%

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

      6. *-commutative 47.7%

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

      7. distribute-rgt-neg-in 47.7%

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

      8. metadata-eval 47.7%

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

   3. Simplified 47.7%

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

      1. fma-udef 47.5%

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

      2. *-commutative 47.5%

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

   5. Applied egg-rr 47.5%

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

   6. Taylor expanded in b around inf 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unpow2 87.9%

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

      4. associate-*r* 87.9%

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

      5. unsub-neg 87.9%

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

      6. associate-*r/ 87.9%

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

      7. neg-mul-1 87.9%

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

      8. associate-*r* 87.9%

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

      9. unpow2 87.9%

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

      10. *-commutative 87.9%

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

      11. unpow2 87.9%

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

   8. Simplified 87.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.55:\\ \;\;\;\;\left(\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 11: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.3)
   (/ (- (sqrt (+ (* b b) (* -4.0 (* a c)))) b) (* a 2.0))
   (- (/ (- c) b) (/ (* a (* c c)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.3) {
		tmp = (sqrt(((b * b) + (-4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * (c * c)) / pow(b, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.3d0) then
        tmp = (sqrt(((b * b) + ((-4.0d0) * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a * (c * c)) / (b ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.3) {
		tmp = (Math.sqrt(((b * b) + (-4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a * (c * c)) / Math.pow(b, 3.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.3:
		tmp = (math.sqrt(((b * b) + (-4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a * (c * c)) / math.pow(b, 3.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.3)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a * Float64(c * c)) / (b ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.3)
		tmp = (sqrt(((b * b) + (-4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((a * (c * c)) / (b ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.3], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2999999999999998

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. +-commutative 82.2%

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

      3. unsub-neg 82.2%

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

      4. fma-neg 82.3%

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

      5. associate-*l* 82.3%

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

      6. *-commutative 82.3%

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

      7. distribute-rgt-neg-in 82.3%

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

      8. metadata-eval 82.3%

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

   3. Simplified 82.3%

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

      1. fma-udef 82.2%

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

      2. *-commutative 82.2%

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

   5. Applied egg-rr 82.2%

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

   if 2.2999999999999998 < b

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

      2. +-commutative 47.5%

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

      3. unsub-neg 47.5%

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

      4. fma-neg 47.7%

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

      5. associate-*l* 47.7%

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

      6. *-commutative 47.7%

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

      7. distribute-rgt-neg-in 47.7%

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

      8. metadata-eval 47.7%

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

   3. Simplified 47.7%

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

      1. fma-udef 47.5%

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

      2. *-commutative 47.5%

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

   5. Applied egg-rr 47.5%

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

   6. Taylor expanded in b around inf 87.9%

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

      1. +-commutative 87.9%

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

      2. mul-1-neg 87.9%

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

      3. unpow2 87.9%

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

      4. associate-*r* 87.9%

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

      5. unsub-neg 87.9%

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

      6. associate-*r/ 87.9%

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

      7. neg-mul-1 87.9%

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

      8. associate-*r* 87.9%

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

      9. unpow2 87.9%

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

      10. *-commutative 87.9%

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

      11. unpow2 87.9%

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

   8. Simplified 87.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.3:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\\ \end{array} \]

Alternative 12: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-c / b) - ((a * (c * c)) / (b ** 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. *-commutative 54.3%

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

   2. +-commutative 54.3%

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

   3. unsub-neg 54.3%

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

   4. fma-neg 54.4%

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

   5. associate-*l* 54.4%

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

   6. *-commutative 54.4%

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

   7. distribute-rgt-neg-in 54.4%

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

   8. metadata-eval 54.4%

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

3. Simplified 54.4%

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

   1. fma-udef 54.3%

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

   2. *-commutative 54.3%

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

5. Applied egg-rr 54.3%

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

6. Taylor expanded in b around inf 82.6%

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

   1. +-commutative 82.6%

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

   2. mul-1-neg 82.6%

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

   3. unpow2 82.6%

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

   4. associate-*r* 82.6%

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

   5. unsub-neg 82.6%

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

   6. associate-*r/ 82.6%

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

   7. neg-mul-1 82.6%

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

   8. associate-*r* 82.6%

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

   9. unpow2 82.6%

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

   10. *-commutative 82.6%

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

   11. unpow2 82.6%

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

8. Simplified 82.6%

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

9. Final simplification 82.6%

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

Alternative 13: 64.3% 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 54.3%

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

   1. neg-sub0 54.3%

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

   2. associate-+l- 54.3%

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

   3. sub0-neg 54.3%

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

   4. neg-mul-1 54.3%

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

   5. associate-*l/ 54.3%

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

   6. *-commutative 54.3%

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

   7. associate-/r* 54.3%

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

   8. /-rgt-identity 54.3%

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

   9. metadata-eval 54.3%

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

3. Simplified 54.3%

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

4. Taylor expanded in b around inf 65.2%

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

   1. associate-*r/ 65.2%

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

   2. neg-mul-1 65.2%

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

6. Simplified 65.2%

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

7. Final simplification 65.2%

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

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 54.3%

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

   1. add-log-exp 48.2%

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

   2. neg-mul-1 48.2%

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

   3. fma-def 48.2%

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

   4. *-commutative 48.2%

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

   5. *-commutative 48.2%

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

   6. *-commutative 48.2%

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

3. Applied egg-rr 48.2%

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

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 2023217 
(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)))