Quadratic roots, narrow range

Percentage Accurate: 55.7% → 99.3%

Time: 28.1s

Alternatives: 13

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: 55.7% 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: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\\ 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\mathsf{fma}\left(t\_0, t\_0, b\right)}}{a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (fma a (* c -4.0) (pow b 2.0)) 0.25)))
   (* 0.5 (/ (/ (+ (* (pow b 2.0) 0.0) (* a (* c -4.0))) (fma t_0 t_0 b)) a))))

C

double code(double a, double b, double c) {
	double t_0 = pow(fma(a, (c * -4.0), pow(b, 2.0)), 0.25);
	return 0.5 * ((((pow(b, 2.0) * 0.0) + (a * (c * -4.0))) / fma(t_0, t_0, b)) / a);
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(c * -4.0), (b ^ 2.0)) ^ 0.25
	return Float64(0.5 * Float64(Float64(Float64(Float64((b ^ 2.0) * 0.0) + Float64(a * Float64(c * -4.0))) / fma(t_0, t_0, b)) / a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * N[(c * -4.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]}, N[(0.5 * N[(N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] * 0.0), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.9%

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

3. Simplified 54.9%

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

   1. frac-2neg 54.9%

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

   2. div-inv 54.9%

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

   3. sub-neg 54.9%

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

   4. distribute-neg-in 54.9%

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

   5. pow2 54.9%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 54.9%

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

   13. sqr-neg 54.9%

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

   14. sqrt-prod 53.7%

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

   15. add-sqr-sqrt 54.9%

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

   16. distribute-rgt-neg-in 54.9%

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

   17. metadata-eval 54.9%

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

6. Applied egg-rr 54.9%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. unpow2 54.8%

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

8. Applied egg-rr 54.8%

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

   1. unpow2 54.8%

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

   2. sqr-neg 54.8%

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

   3. rem-square-sqrt 56.5%

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

   4. fma-define 56.5%

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

   5. associate-*r* 56.5%

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

   6. fma-define 56.5%

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

   7. sub-neg 56.5%

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

   8. distribute-neg-out 56.5%

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

   9. fma-define 56.5%

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

   10. associate-*r* 56.5%

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

   11. fma-define 56.5%

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

10. Simplified 56.5%

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

    1. un-div-inv 56.4%

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

    2. distribute-frac-neg2 56.4%

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

    3. +-commutative 56.4%

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

12. Applied egg-rr 56.4%

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

    1. neg-mul-1 56.4%

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

    2. *-commutative 56.4%

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

    3. times-frac 56.4%

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

    4. metadata-eval 56.4%

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

14. Simplified 99.3%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}{a}} \]
15. Step-by-step derivation

    1. +-commutative 99.3%

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

    2. fma-define 99.3%

       \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + {b}^{2}}} + b}}{a} \]

    3. add-sqr-sqrt 99.2%

       \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\color{blue}{\sqrt{\sqrt{a \cdot \left(c \cdot -4\right) + {b}^{2}}} \cdot \sqrt{\sqrt{a \cdot \left(c \cdot -4\right) + {b}^{2}}}} + b}}{a} \]

    4. fma-define 99.3%

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

    5. pow1/2 99.3%

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

    6. sqrt-pow1 99.3%

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

    7. fma-define 99.3%

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

    8. metadata-eval 99.3%

       \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{a \cdot \left(c \cdot -4\right) + {b}^{2}}}, b\right)}}{a} \]

    9. pow1/2 99.3%

       \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}, \sqrt{\color{blue}{{\left(a \cdot \left(c \cdot -4\right) + {b}^{2}\right)}^{0.5}}}, b\right)}}{a} \]

    10. sqrt-pow1 99.3%

        \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}, \color{blue}{{\left(a \cdot \left(c \cdot -4\right) + {b}^{2}\right)}^{\left(\frac{0.5}{2}\right)}}, b\right)}}{a} \]

    11. fma-define 99.3%

        \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}, {\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}}^{\left(\frac{0.5}{2}\right)}, b\right)}}{a} \]

    12. metadata-eval 99.3%

        \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}, b\right)}}{a} \]

16. Applied egg-rr 99.3%

    \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}, b\right)}}}{a} \]

17. Final simplification 99.3%

    \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\mathsf{fma}\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}, b\right)}}{a} \]
18. Add Preprocessing

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  0.5
  (/
   (/
    (+ (* (pow b 2.0) 0.0) (* a (* c -4.0)))
    (+ b (sqrt (fma a (* c -4.0) (pow b 2.0)))))
   a)))

C

double code(double a, double b, double c) {
	return 0.5 * ((((pow(b, 2.0) * 0.0) + (a * (c * -4.0))) / (b + sqrt(fma(a, (c * -4.0), pow(b, 2.0))))) / a);
}

Julia

function code(a, b, c)
	return Float64(0.5 * Float64(Float64(Float64(Float64((b ^ 2.0) * 0.0) + Float64(a * Float64(c * -4.0))) / Float64(b + sqrt(fma(a, Float64(c * -4.0), (b ^ 2.0))))) / a))
end

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Simplified 54.9%

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

   1. frac-2neg 54.9%

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

   2. div-inv 54.9%

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

   3. sub-neg 54.9%

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

   4. distribute-neg-in 54.9%

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

   5. pow2 54.9%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 54.9%

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

   13. sqr-neg 54.9%

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

   14. sqrt-prod 53.7%

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

   15. add-sqr-sqrt 54.9%

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

   16. distribute-rgt-neg-in 54.9%

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

   17. metadata-eval 54.9%

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

6. Applied egg-rr 54.9%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. unpow2 54.8%

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

8. Applied egg-rr 54.8%

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

   1. unpow2 54.8%

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

   2. sqr-neg 54.8%

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

   3. rem-square-sqrt 56.5%

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

   4. fma-define 56.5%

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

   5. associate-*r* 56.5%

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

   6. fma-define 56.5%

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

   7. sub-neg 56.5%

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

   8. distribute-neg-out 56.5%

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

   9. fma-define 56.5%

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

   10. associate-*r* 56.5%

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

   11. fma-define 56.5%

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

10. Simplified 56.5%

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

    1. un-div-inv 56.4%

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

    2. distribute-frac-neg2 56.4%

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

    3. +-commutative 56.4%

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

12. Applied egg-rr 56.4%

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

    1. neg-mul-1 56.4%

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

    2. *-commutative 56.4%

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

    3. times-frac 56.4%

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

    4. metadata-eval 56.4%

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

14. Simplified 99.3%

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

15. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{b + \sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)}}}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  0.5
  (/
   (/
    (+ (* (pow b 2.0) 0.0) (* a (* c -4.0)))
    (+ b (sqrt (* c (+ (* a -4.0) (/ (pow b 2.0) c))))))
   a)))

C

double code(double a, double b, double c) {
	return 0.5 * ((((pow(b, 2.0) * 0.0) + (a * (c * -4.0))) / (b + sqrt((c * ((a * -4.0) + (pow(b, 2.0) / c)))))) / 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 = 0.5d0 * (((((b ** 2.0d0) * 0.0d0) + (a * (c * (-4.0d0)))) / (b + sqrt((c * ((a * (-4.0d0)) + ((b ** 2.0d0) / c)))))) / a)
end function

Java

public static double code(double a, double b, double c) {
	return 0.5 * ((((Math.pow(b, 2.0) * 0.0) + (a * (c * -4.0))) / (b + Math.sqrt((c * ((a * -4.0) + (Math.pow(b, 2.0) / c)))))) / a);
}

Python

def code(a, b, c):
	return 0.5 * ((((math.pow(b, 2.0) * 0.0) + (a * (c * -4.0))) / (b + math.sqrt((c * ((a * -4.0) + (math.pow(b, 2.0) / c)))))) / a)

Julia

function code(a, b, c)
	return Float64(0.5 * Float64(Float64(Float64(Float64((b ^ 2.0) * 0.0) + Float64(a * Float64(c * -4.0))) / Float64(b + sqrt(Float64(c * Float64(Float64(a * -4.0) + Float64((b ^ 2.0) / c)))))) / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.5 * (((((b ^ 2.0) * 0.0) + (a * (c * -4.0))) / (b + sqrt((c * ((a * -4.0) + ((b ^ 2.0) / c)))))) / a);
end

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Simplified 54.9%

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

   1. frac-2neg 54.9%

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

   2. div-inv 54.9%

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

   3. sub-neg 54.9%

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

   4. distribute-neg-in 54.9%

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

   5. pow2 54.9%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 54.9%

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

   13. sqr-neg 54.9%

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

   14. sqrt-prod 53.7%

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

   15. add-sqr-sqrt 54.9%

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

   16. distribute-rgt-neg-in 54.9%

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

   17. metadata-eval 54.9%

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

6. Applied egg-rr 54.9%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. unpow2 54.8%

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

8. Applied egg-rr 54.8%

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

   1. unpow2 54.8%

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

   2. sqr-neg 54.8%

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

   3. rem-square-sqrt 56.5%

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

   4. fma-define 56.5%

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

   5. associate-*r* 56.5%

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

   6. fma-define 56.5%

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

   7. sub-neg 56.5%

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

   8. distribute-neg-out 56.5%

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

   9. fma-define 56.5%

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

   10. associate-*r* 56.5%

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

   11. fma-define 56.5%

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

10. Simplified 56.5%

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

    1. un-div-inv 56.4%

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

    2. distribute-frac-neg2 56.4%

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

    3. +-commutative 56.4%

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

12. Applied egg-rr 56.4%

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

    1. neg-mul-1 56.4%

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

    2. *-commutative 56.4%

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

    3. times-frac 56.4%

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

    4. metadata-eval 56.4%

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

14. Simplified 99.3%

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

15. Taylor expanded in c around inf 99.3%

    \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{b + \sqrt{\color{blue}{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)}}}}{a} \]

16. Final simplification 99.3%

    \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{b + \sqrt{c \cdot \left(a \cdot -4 + \frac{{b}^{2}}{c}\right)}}}{a} \]
17. Add Preprocessing

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -4\right)\\ 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + t\_0}{b + \sqrt{{b}^{2} + t\_0}}}{a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c -4.0))))
   (*
    0.5
    (/ (/ (+ (* (pow b 2.0) 0.0) t_0) (+ b (sqrt (+ (pow b 2.0) t_0)))) a))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * -4.0);
	return 0.5 * ((((pow(b, 2.0) * 0.0) + t_0) / (b + sqrt((pow(b, 2.0) + t_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
    real(8) :: t_0
    t_0 = a * (c * (-4.0d0))
    code = 0.5d0 * (((((b ** 2.0d0) * 0.0d0) + t_0) / (b + sqrt(((b ** 2.0d0) + t_0)))) / a)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * -4.0);
	return 0.5 * ((((Math.pow(b, 2.0) * 0.0) + t_0) / (b + Math.sqrt((Math.pow(b, 2.0) + t_0)))) / a);
}

Python

def code(a, b, c):
	t_0 = a * (c * -4.0)
	return 0.5 * ((((math.pow(b, 2.0) * 0.0) + t_0) / (b + math.sqrt((math.pow(b, 2.0) + t_0)))) / a)

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -4.0))
	return Float64(0.5 * Float64(Float64(Float64(Float64((b ^ 2.0) * 0.0) + t_0) / Float64(b + sqrt(Float64((b ^ 2.0) + t_0)))) / a))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (c * -4.0);
	tmp = 0.5 * (((((b ^ 2.0) * 0.0) + t_0) / (b + sqrt(((b ^ 2.0) + t_0)))) / a);
end

Wolfram

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

Derivation

1. Initial program 54.9%

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

3. Simplified 54.9%

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

   1. frac-2neg 54.9%

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

   2. div-inv 54.9%

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

   3. sub-neg 54.9%

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

   4. distribute-neg-in 54.9%

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

   5. pow2 54.9%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 54.9%

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

   13. sqr-neg 54.9%

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

   14. sqrt-prod 53.7%

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

   15. add-sqr-sqrt 54.9%

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

   16. distribute-rgt-neg-in 54.9%

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

   17. metadata-eval 54.9%

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

6. Applied egg-rr 54.9%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. unpow2 54.8%

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

8. Applied egg-rr 54.8%

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

   1. unpow2 54.8%

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

   2. sqr-neg 54.8%

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

   3. rem-square-sqrt 56.5%

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

   4. fma-define 56.5%

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

   5. associate-*r* 56.5%

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

   6. fma-define 56.5%

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

   7. sub-neg 56.5%

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

   8. distribute-neg-out 56.5%

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

   9. fma-define 56.5%

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

   10. associate-*r* 56.5%

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

   11. fma-define 56.5%

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

10. Simplified 56.5%

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

    1. un-div-inv 56.4%

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

    2. distribute-frac-neg2 56.4%

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

    3. +-commutative 56.4%

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

12. Applied egg-rr 56.4%

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

    1. neg-mul-1 56.4%

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

    2. *-commutative 56.4%

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

    3. times-frac 56.4%

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

    4. metadata-eval 56.4%

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

14. Simplified 99.3%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}}}{a}} \]
15. Step-by-step derivation

    1. fma-undefine 99.3%

       \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + {b}^{2}}}}}{a} \]

16. Applied egg-rr 99.3%

    \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + {b}^{2}}}}}{a} \]

17. Final simplification 99.3%

    \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{b + \sqrt{{b}^{2} + a \cdot \left(c \cdot -4\right)}}}{a} \]
18. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}} \cdot \frac{-1}{a \cdot -2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ (* a (* c -4.0)) (+ b (sqrt (fma (* a c) -4.0 (pow b 2.0)))))
  (/ -1.0 (* a -2.0))))

C

double code(double a, double b, double c) {
	return ((a * (c * -4.0)) / (b + sqrt(fma((a * c), -4.0, pow(b, 2.0))))) * (-1.0 / (a * -2.0));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(c * -4.0)) / Float64(b + sqrt(fma(Float64(a * c), -4.0, (b ^ 2.0))))) * Float64(-1.0 / Float64(a * -2.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

3. Simplified 54.9%

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

   1. frac-2neg 54.9%

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

   2. div-inv 54.9%

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

   3. sub-neg 54.9%

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

   4. distribute-neg-in 54.9%

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

   5. pow2 54.9%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 54.9%

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

   13. sqr-neg 54.9%

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

   14. sqrt-prod 53.7%

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

   15. add-sqr-sqrt 54.9%

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

   16. distribute-rgt-neg-in 54.9%

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

   17. metadata-eval 54.9%

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

6. Applied egg-rr 54.9%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. unpow2 54.8%

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

8. Applied egg-rr 54.8%

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

   1. unpow2 54.8%

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

   2. sqr-neg 54.8%

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

   3. rem-square-sqrt 56.5%

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

   4. fma-define 56.5%

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

   5. associate-*r* 56.5%

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

   6. fma-define 56.5%

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

   7. sub-neg 56.5%

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

   8. distribute-neg-out 56.5%

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

   9. fma-define 56.5%

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

   10. associate-*r* 56.5%

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

   11. fma-define 56.5%

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

10. Simplified 56.5%

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

11. Taylor expanded in a around 0 99.2%

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

    1. *-commutative 99.2%

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

    2. associate-*l* 99.2%

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

13. Simplified 99.2%

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

14. Final simplification 99.2%

    \[\leadsto \frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, {b}^{2}\right)}} \cdot \frac{-1}{a \cdot -2} \]
15. Add Preprocessing

Alternative 6: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.27:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{c} + a \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.27)
   (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* 2.0 a))
   (*
    (/ 1.0 (* a -2.0))
    (/
     1.0
     (/
      (+
       (* 0.5 (/ b c))
       (* a (+ (* -0.5 (/ (* a c) (pow b 3.0))) (* 0.5 (/ -1.0 b)))))
      a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.27) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (a * ((-0.5 * ((a * c) / pow(b, 3.0))) + (0.5 * (-1.0 / b))))) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.27)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(b / c)) + Float64(a * Float64(Float64(-0.5 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(0.5 * Float64(-1.0 / b))))) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.27], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.27000000000000002

   1. Initial program 87.0%

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

   3. Simplified 87.1%

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

   if 0.27000000000000002 < b

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. frac-2neg 50.2%

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

      2. div-inv 50.2%

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

      3. sub-neg 50.2%

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

      4. distribute-neg-in 50.2%

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

      5. pow2 50.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 50.2%

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

      13. sqr-neg 50.2%

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

      14. sqrt-prod 49.0%

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

      15. add-sqr-sqrt 50.2%

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

      16. distribute-rgt-neg-in 50.2%

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

      17. metadata-eval 50.2%

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

   6. Applied egg-rr 50.2%

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

      1. flip-+ 50.0%

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

      2. pow2 50.0%

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

      3. unpow2 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow2 50.0%

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

      2. sqr-neg 50.0%

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

      3. rem-square-sqrt 51.8%

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

      4. fma-define 51.8%

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

      5. associate-*r* 51.8%

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

      6. fma-define 51.8%

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

      7. sub-neg 51.8%

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

      8. distribute-neg-out 51.8%

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

      9. fma-define 51.7%

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

      10. associate-*r* 51.7%

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

      11. fma-define 51.7%

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

   10. Simplified 51.7%

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

       1. clear-num 51.7%

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

       2. inv-pow 51.7%

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

       3. +-commutative 51.7%

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

   12. Applied egg-rr 51.7%

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

       1. unpow-1 51.7%

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

       2. distribute-frac-neg 51.7%

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

       3. distribute-neg-frac2 51.7%

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

       4. fma-define 51.7%

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

       5. associate-*l* 51.7%

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

       6. *-commutative 51.7%

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

       7. fma-define 51.7%

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

       8. *-commutative 51.7%

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

       9. sub-neg 51.7%

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

       10. +-commutative 51.7%

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

       11. fma-define 51.7%

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

       12. +-commutative 51.7%

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

       13. *-commutative 51.7%

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

       14. associate-+r+ 99.1%

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

   14. Simplified 99.1%

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

   15. Taylor expanded in a around 0 90.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{0.5 \cdot \frac{b}{c} + a \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)}{a}}} \cdot \frac{1}{a \cdot -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.27:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{c} + a \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{c} + a \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.6)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* 2.0 a))
   (*
    (/ 1.0 (* a -2.0))
    (/
     1.0
     (/
      (+
       (* 0.5 (/ b c))
       (* a (+ (* -0.5 (/ (* a c) (pow b 3.0))) (* 0.5 (/ -1.0 b)))))
      a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.6) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (a * ((-0.5 * ((a * c) / pow(b, 3.0))) + (0.5 * (-1.0 / b))))) / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.6)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(b / c)) + Float64(a * Float64(Float64(-0.5 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(0.5 * Float64(-1.0 / b))))) / a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.6], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.599999999999999978

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

   3. Simplified 87.2%

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

   if 0.599999999999999978 < b

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. frac-2neg 50.2%

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

      2. div-inv 50.2%

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

      3. sub-neg 50.2%

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

      4. distribute-neg-in 50.2%

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

      5. pow2 50.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 50.2%

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

      13. sqr-neg 50.2%

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

      14. sqrt-prod 49.0%

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

      15. add-sqr-sqrt 50.2%

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

      16. distribute-rgt-neg-in 50.2%

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

      17. metadata-eval 50.2%

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

   6. Applied egg-rr 50.2%

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

      1. flip-+ 50.0%

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

      2. pow2 50.0%

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

      3. unpow2 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow2 50.0%

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

      2. sqr-neg 50.0%

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

      3. rem-square-sqrt 51.8%

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

      4. fma-define 51.8%

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

      5. associate-*r* 51.8%

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

      6. fma-define 51.8%

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

      7. sub-neg 51.8%

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

      8. distribute-neg-out 51.8%

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

      9. fma-define 51.7%

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

      10. associate-*r* 51.7%

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

      11. fma-define 51.7%

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

   10. Simplified 51.7%

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

       1. clear-num 51.7%

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

       2. inv-pow 51.7%

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

       3. +-commutative 51.7%

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

   12. Applied egg-rr 51.7%

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

       1. unpow-1 51.7%

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

       2. distribute-frac-neg 51.7%

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

       3. distribute-neg-frac2 51.7%

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

       4. fma-define 51.7%

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

       5. associate-*l* 51.7%

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

       6. *-commutative 51.7%

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

       7. fma-define 51.7%

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

       8. *-commutative 51.7%

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

       9. sub-neg 51.7%

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

       10. +-commutative 51.7%

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

       11. fma-define 51.7%

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

       12. +-commutative 51.7%

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

       13. *-commutative 51.7%

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

       14. associate-+r+ 99.1%

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

   14. Simplified 99.1%

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

   15. Taylor expanded in a around 0 90.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{0.5 \cdot \frac{b}{c} + a \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)}{a}}} \cdot \frac{1}{a \cdot -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{c} + a \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.31:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{a} + c \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.31)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (*
    (/ 1.0 (* a -2.0))
    (/
     1.0
     (/
      (+
       (* 0.5 (/ b a))
       (* c (+ (* -0.5 (/ (* a c) (pow b 3.0))) (* 0.5 (/ -1.0 b)))))
      c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.31) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / a)) + (c * ((-0.5 * ((a * c) / pow(b, 3.0))) + (0.5 * (-1.0 / b))))) / c));
	}
	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 <= 0.31d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = (1.0d0 / (a * (-2.0d0))) * (1.0d0 / (((0.5d0 * (b / a)) + (c * (((-0.5d0) * ((a * c) / (b ** 3.0d0))) + (0.5d0 * ((-1.0d0) / b))))) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.31) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / a)) + (c * ((-0.5 * ((a * c) / Math.pow(b, 3.0))) + (0.5 * (-1.0 / b))))) / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.31:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / a)) + (c * ((-0.5 * ((a * c) / math.pow(b, 3.0))) + (0.5 * (-1.0 / b))))) / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.31)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(b / a)) + Float64(c * Float64(Float64(-0.5 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(0.5 * Float64(-1.0 / b))))) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.31)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / a)) + (c * ((-0.5 * ((a * c) / (b ^ 3.0))) + (0.5 * (-1.0 / b))))) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.31], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(-0.5 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.309999999999999998

   1. Initial program 87.0%

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

   if 0.309999999999999998 < b

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. frac-2neg 50.2%

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

      2. div-inv 50.2%

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

      3. sub-neg 50.2%

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

      4. distribute-neg-in 50.2%

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

      5. pow2 50.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 50.2%

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

      13. sqr-neg 50.2%

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

      14. sqrt-prod 49.0%

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

      15. add-sqr-sqrt 50.2%

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

      16. distribute-rgt-neg-in 50.2%

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

      17. metadata-eval 50.2%

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

   6. Applied egg-rr 50.2%

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

      1. flip-+ 50.0%

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

      2. pow2 50.0%

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

      3. unpow2 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow2 50.0%

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

      2. sqr-neg 50.0%

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

      3. rem-square-sqrt 51.8%

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

      4. fma-define 51.8%

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

      5. associate-*r* 51.8%

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

      6. fma-define 51.8%

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

      7. sub-neg 51.8%

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

      8. distribute-neg-out 51.8%

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

      9. fma-define 51.7%

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

      10. associate-*r* 51.7%

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

      11. fma-define 51.7%

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

   10. Simplified 51.7%

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

       1. clear-num 51.7%

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

       2. inv-pow 51.7%

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

       3. +-commutative 51.7%

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

   12. Applied egg-rr 51.7%

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

       1. unpow-1 51.7%

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

       2. distribute-frac-neg 51.7%

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

       3. distribute-neg-frac2 51.7%

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

       4. fma-define 51.7%

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

       5. associate-*l* 51.7%

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

       6. *-commutative 51.7%

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

       7. fma-define 51.7%

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

       8. *-commutative 51.7%

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

       9. sub-neg 51.7%

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

       10. +-commutative 51.7%

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

       11. fma-define 51.7%

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

       12. +-commutative 51.7%

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

       13. *-commutative 51.7%

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

       14. associate-+r+ 99.1%

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

   14. Simplified 99.1%

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

   15. Taylor expanded in c around 0 90.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{0.5 \cdot \frac{b}{a} + c \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)}{c}}} \cdot \frac{1}{a \cdot -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.31:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{a} + c \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.27:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{c} + a \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.27)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (*
    (/ 1.0 (* a -2.0))
    (/
     1.0
     (/
      (+
       (* 0.5 (/ b c))
       (* a (+ (* -0.5 (/ (* a c) (pow b 3.0))) (* 0.5 (/ -1.0 b)))))
      a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.27) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (a * ((-0.5 * ((a * c) / pow(b, 3.0))) + (0.5 * (-1.0 / b))))) / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.27d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = (1.0d0 / (a * (-2.0d0))) * (1.0d0 / (((0.5d0 * (b / c)) + (a * (((-0.5d0) * ((a * c) / (b ** 3.0d0))) + (0.5d0 * ((-1.0d0) / b))))) / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.27) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (a * ((-0.5 * ((a * c) / Math.pow(b, 3.0))) + (0.5 * (-1.0 / b))))) / a));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.27:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (a * ((-0.5 * ((a * c) / math.pow(b, 3.0))) + (0.5 * (-1.0 / b))))) / a))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.27)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(b / c)) + Float64(a * Float64(Float64(-0.5 * Float64(Float64(a * c) / (b ^ 3.0))) + Float64(0.5 * Float64(-1.0 / b))))) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.27)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (a * ((-0.5 * ((a * c) / (b ^ 3.0))) + (0.5 * (-1.0 / b))))) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.27], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.27000000000000002

   1. Initial program 87.0%

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

   if 0.27000000000000002 < b

   1. Initial program 50.2%

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

   3. Simplified 50.2%

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

      1. frac-2neg 50.2%

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

      2. div-inv 50.2%

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

      3. sub-neg 50.2%

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

      4. distribute-neg-in 50.2%

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

      5. pow2 50.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 50.2%

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

      13. sqr-neg 50.2%

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

      14. sqrt-prod 49.0%

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

      15. add-sqr-sqrt 50.2%

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

      16. distribute-rgt-neg-in 50.2%

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

      17. metadata-eval 50.2%

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

   6. Applied egg-rr 50.2%

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

      1. flip-+ 50.0%

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

      2. pow2 50.0%

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

      3. unpow2 50.0%

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

   8. Applied egg-rr 50.0%

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

      1. unpow2 50.0%

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

      2. sqr-neg 50.0%

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

      3. rem-square-sqrt 51.8%

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

      4. fma-define 51.8%

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

      5. associate-*r* 51.8%

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

      6. fma-define 51.8%

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

      7. sub-neg 51.8%

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

      8. distribute-neg-out 51.8%

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

      9. fma-define 51.7%

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

      10. associate-*r* 51.7%

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

      11. fma-define 51.7%

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

   10. Simplified 51.7%

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

       1. clear-num 51.7%

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

       2. inv-pow 51.7%

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

       3. +-commutative 51.7%

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

   12. Applied egg-rr 51.7%

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

       1. unpow-1 51.7%

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

       2. distribute-frac-neg 51.7%

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

       3. distribute-neg-frac2 51.7%

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

       4. fma-define 51.7%

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

       5. associate-*l* 51.7%

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

       6. *-commutative 51.7%

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

       7. fma-define 51.7%

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

       8. *-commutative 51.7%

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

       9. sub-neg 51.7%

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

       10. +-commutative 51.7%

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

       11. fma-define 51.7%

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

       12. +-commutative 51.7%

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

       13. *-commutative 51.7%

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

       14. associate-+r+ 99.1%

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

   14. Simplified 99.1%

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

   15. Taylor expanded in a around 0 90.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{0.5 \cdot \frac{b}{c} + a \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)}{a}}} \cdot \frac{1}{a \cdot -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.27:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{c} + a \cdot \left(-0.5 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{-1}{b}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{-2 \cdot \frac{a \cdot c}{b} + b \cdot 2}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.3)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (*
    0.5
    (/
     (/
      (+ (* (pow b 2.0) 0.0) (* a (* c -4.0)))
      (+ (* -2.0 (/ (* a c) b)) (* b 2.0)))
     a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.3) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = 0.5 * ((((pow(b, 2.0) * 0.0) + (a * (c * -4.0))) / ((-2.0 * ((a * c) / b)) + (b * 2.0))) / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.3d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = 0.5d0 * (((((b ** 2.0d0) * 0.0d0) + (a * (c * (-4.0d0)))) / (((-2.0d0) * ((a * c) / b)) + (b * 2.0d0))) / a)
    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) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = 0.5 * ((((Math.pow(b, 2.0) * 0.0) + (a * (c * -4.0))) / ((-2.0 * ((a * c) / b)) + (b * 2.0))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.3:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = 0.5 * ((((math.pow(b, 2.0) * 0.0) + (a * (c * -4.0))) / ((-2.0 * ((a * c) / b)) + (b * 2.0))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.3)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(Float64((b ^ 2.0) * 0.0) + Float64(a * Float64(c * -4.0))) / Float64(Float64(-2.0 * Float64(Float64(a * c) / b)) + Float64(b * 2.0))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.3)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = 0.5 * (((((b ^ 2.0) * 0.0) + (a * (c * -4.0))) / ((-2.0 * ((a * c) / b)) + (b * 2.0))) / a);
	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[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(N[(N[Power[b, 2.0], $MachinePrecision] * 0.0), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2999999999999998

   1. Initial program 85.8%

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

   if 2.2999999999999998 < b

   1. Initial program 49.4%

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

   3. Simplified 49.4%

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

      1. frac-2neg 49.4%

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

      2. div-inv 49.4%

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

      3. sub-neg 49.4%

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

      4. distribute-neg-in 49.4%

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

      5. pow2 49.4%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 49.4%

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

      13. sqr-neg 49.4%

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

      14. sqrt-prod 48.2%

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

      15. add-sqr-sqrt 49.4%

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

      16. distribute-rgt-neg-in 49.4%

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

      17. metadata-eval 49.4%

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

   6. Applied egg-rr 49.4%

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

      1. flip-+ 49.3%

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

      2. pow2 49.3%

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

      3. unpow2 49.3%

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

   8. Applied egg-rr 49.3%

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

      1. unpow2 49.3%

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

      2. sqr-neg 49.3%

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

      3. rem-square-sqrt 51.0%

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

      4. fma-define 51.0%

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

      5. associate-*r* 51.0%

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

      6. fma-define 51.0%

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

      7. sub-neg 51.0%

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

      8. distribute-neg-out 51.0%

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

      9. fma-define 51.0%

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

      10. associate-*r* 51.0%

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

      11. fma-define 51.0%

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

   10. Simplified 51.0%

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

       1. un-div-inv 51.0%

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

       2. distribute-frac-neg2 51.0%

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

       3. +-commutative 51.0%

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

   12. Applied egg-rr 51.0%

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

       1. neg-mul-1 51.0%

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

       2. *-commutative 51.0%

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

       3. times-frac 51.0%

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

       4. metadata-eval 51.0%

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

   14. Simplified 99.3%

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

   15. Taylor expanded in a around 0 85.9%

       \[\leadsto 0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{\color{blue}{-2 \cdot \frac{a \cdot c}{b} + 2 \cdot b}}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.3:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{{b}^{2} \cdot 0 + a \cdot \left(c \cdot -4\right)}{-2 \cdot \frac{a \cdot c}{b} + b \cdot 2}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{a} + -0.5 \cdot \frac{c}{b}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.3)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
   (* (/ 1.0 (* a -2.0)) (/ 1.0 (/ (+ (* 0.5 (/ b a)) (* -0.5 (/ c b))) c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.3) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / a)) + (-0.5 * (c / b))) / c));
	}
	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) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = (1.0d0 / (a * (-2.0d0))) * (1.0d0 / (((0.5d0 * (b / a)) + ((-0.5d0) * (c / b))) / c))
    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) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / a)) + (-0.5 * (c / b))) / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.3:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / a)) + (-0.5 * (c / b))) / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.3)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(b / a)) + Float64(-0.5 * Float64(c / b))) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.3)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / a)) + (-0.5 * (c / b))) / c));
	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[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2999999999999998

   1. Initial program 85.8%

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

   if 2.2999999999999998 < b

   1. Initial program 49.4%

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

   3. Simplified 49.4%

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

      1. frac-2neg 49.4%

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

      2. div-inv 49.4%

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

      3. sub-neg 49.4%

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

      4. distribute-neg-in 49.4%

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

      5. pow2 49.4%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.6%

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

      8. sqr-neg 1.6%

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

      9. sqrt-prod 1.6%

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

      10. add-sqr-sqrt 1.6%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 49.4%

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

      13. sqr-neg 49.4%

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

      14. sqrt-prod 48.2%

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

      15. add-sqr-sqrt 49.4%

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

      16. distribute-rgt-neg-in 49.4%

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

      17. metadata-eval 49.4%

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

   6. Applied egg-rr 49.4%

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

      1. flip-+ 49.3%

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

      2. pow2 49.3%

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

      3. unpow2 49.3%

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

   8. Applied egg-rr 49.3%

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

      1. unpow2 49.3%

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

      2. sqr-neg 49.3%

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

      3. rem-square-sqrt 51.0%

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

      4. fma-define 51.0%

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

      5. associate-*r* 51.0%

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

      6. fma-define 51.0%

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

      7. sub-neg 51.0%

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

      8. distribute-neg-out 51.0%

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

      9. fma-define 51.0%

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

      10. associate-*r* 51.0%

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

      11. fma-define 51.0%

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

   10. Simplified 51.0%

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

       1. clear-num 51.0%

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

       2. inv-pow 51.0%

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

       3. +-commutative 51.0%

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

   12. Applied egg-rr 51.0%

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

       1. unpow-1 51.0%

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

       2. distribute-frac-neg 51.0%

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

       3. distribute-neg-frac2 51.0%

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

       4. fma-define 51.0%

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

       5. associate-*l* 51.0%

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

       6. *-commutative 51.0%

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

       7. fma-define 51.0%

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

       8. *-commutative 51.0%

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

       9. sub-neg 51.0%

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

       10. +-commutative 51.0%

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

       11. fma-define 51.0%

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

       12. +-commutative 51.0%

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

       13. *-commutative 51.0%

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

       14. associate-+r+ 99.1%

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

   14. Simplified 99.1%

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

   15. Taylor expanded in c around 0 85.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{c}{b} + 0.5 \cdot \frac{b}{a}}{c}}} \cdot \frac{1}{a \cdot -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.3:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{a} + -0.5 \cdot \frac{c}{b}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}{a}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ 1.0 (* a -2.0)) (/ 1.0 (/ (+ (* 0.5 (/ b c)) (* -0.5 (/ a b))) a))))

C

double code(double a, double b, double c) {
	return (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (-0.5 * (a / b))) / 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 = (1.0d0 / (a * (-2.0d0))) * (1.0d0 / (((0.5d0 * (b / c)) + ((-0.5d0) * (a / b))) / a))
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (-0.5 * (a / b))) / a));
}

Python

def code(a, b, c):
	return (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (-0.5 * (a / b))) / a))

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(a * -2.0)) * Float64(1.0 / Float64(Float64(Float64(0.5 * Float64(b / c)) + Float64(-0.5 * Float64(a / b))) / a)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / (a * -2.0)) * (1.0 / (((0.5 * (b / c)) + (-0.5 * (a / b))) / a));
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(a * -2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

3. Simplified 54.9%

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

   1. frac-2neg 54.9%

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

   2. div-inv 54.9%

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

   3. sub-neg 54.9%

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

   4. distribute-neg-in 54.9%

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

   5. pow2 54.9%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 1.6%

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

   8. sqr-neg 1.6%

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

   9. sqrt-prod 1.6%

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

   10. add-sqr-sqrt 1.6%

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

   11. add-sqr-sqrt 0.0%

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

   12. sqrt-unprod 54.9%

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

   13. sqr-neg 54.9%

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

   14. sqrt-prod 53.7%

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

   15. add-sqr-sqrt 54.9%

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

   16. distribute-rgt-neg-in 54.9%

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

   17. metadata-eval 54.9%

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

6. Applied egg-rr 54.9%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. unpow2 54.8%

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

8. Applied egg-rr 54.8%

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

   1. unpow2 54.8%

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

   2. sqr-neg 54.8%

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

   3. rem-square-sqrt 56.5%

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

   4. fma-define 56.5%

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

   5. associate-*r* 56.5%

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

   6. fma-define 56.5%

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

   7. sub-neg 56.5%

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

   8. distribute-neg-out 56.5%

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

   9. fma-define 56.5%

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

   10. associate-*r* 56.5%

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

   11. fma-define 56.5%

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

10. Simplified 56.5%

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

    1. clear-num 56.5%

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

    2. inv-pow 56.5%

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

    3. +-commutative 56.5%

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

12. Applied egg-rr 56.5%

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

    1. unpow-1 56.5%

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

    2. distribute-frac-neg 56.5%

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

    3. distribute-neg-frac2 56.5%

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

    4. fma-define 56.5%

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

    5. associate-*l* 56.5%

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

    6. *-commutative 56.5%

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

    7. fma-define 56.5%

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

    8. *-commutative 56.5%

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

    9. sub-neg 56.5%

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

    10. +-commutative 56.5%

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

    11. fma-define 56.5%

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

    12. +-commutative 56.5%

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

    13. *-commutative 56.5%

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

    14. associate-+r+ 99.1%

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

14. Simplified 99.1%

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

15. Taylor expanded in a around 0 80.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{a}{b} + 0.5 \cdot \frac{b}{c}}{a}}} \cdot \frac{1}{a \cdot -2} \]

16. Final simplification 80.9%

    \[\leadsto \frac{1}{a \cdot -2} \cdot \frac{1}{\frac{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}{a}} \]
17. Add Preprocessing

Alternative 13: 64.2% 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(c / Float64(-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 54.9%

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

   1. *-commutative 54.9%

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

3. Simplified 54.9%

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

5. Taylor expanded in b around inf 64.7%

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

   1. associate-*r/ 64.7%

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

   2. mul-1-neg 64.7%

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

7. Simplified 64.7%

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

8. Final simplification 64.7%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(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)))