Quadratic roots, narrow range

Percentage Accurate: 55.8% → 99.4%

Time: 16.4s

Alternatives: 8

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{-2 \cdot \frac{a \cdot c}{a}}{b + \sqrt{\frac{{b}^{4} - {\left(a \cdot c\right)}^{2} \cdot 16}{\mathsf{fma}\left(a \cdot c, 4, b \cdot b\right)}}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* -2.0 (/ (* a c) a))
  (+
   b
   (sqrt
    (/
     (- (pow b 4.0) (* (pow (* a c) 2.0) 16.0))
     (fma (* a c) 4.0 (* b b)))))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((a * c) / a)) / (b + sqrt(((pow(b, 4.0) - (pow((a * c), 2.0) * 16.0)) / fma((a * c), 4.0, (b * b)))));
}

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64(a * c) / a)) / Float64(b + sqrt(Float64(Float64((b ^ 4.0) - Float64((Float64(a * c) ^ 2.0) * 16.0)) / fma(Float64(a * c), 4.0, Float64(b * b))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(N[Power[b, 4.0], $MachinePrecision] - N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] * 16.0), $MachinePrecision]), $MachinePrecision] / N[(N[(a * c), $MachinePrecision] * 4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.9%

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

3. Simplified 53.9%

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

   1. *-commutative 53.9%

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

   2. metadata-eval 53.9%

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

   3. distribute-lft-neg-in 53.9%

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

   4. distribute-rgt-neg-in 53.9%

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

   5. *-commutative 53.9%

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

   6. fma-neg 53.9%

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

   7. flip-- 53.8%

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

   8. div-sub 53.8%

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

   9. pow2 53.8%

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

   10. pow2 53.8%

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

   11. pow-prod-up 53.5%

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

   12. metadata-eval 53.5%

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

   13. fma-def 53.8%

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

   14. associate-*l* 53.8%

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

   15. pow2 53.8%

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

   16. associate-*l* 53.8%

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

   17. fma-def 53.8%

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

   18. associate-*l* 53.8%

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

5. Applied egg-rr 53.8%

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

   1. fma-def 53.5%

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

   2. +-commutative 53.5%

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

   3. *-commutative 53.5%

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

   4. fma-def 53.5%

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

   5. *-commutative 53.5%

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

   6. *-commutative 53.5%

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

   7. associate-*l* 53.5%

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

   8. fma-def 53.5%

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

   9. +-commutative 53.5%

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

   10. *-commutative 53.5%

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

   11. fma-def 53.5%

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

7. Simplified 53.5%

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

   1. flip-- 53.3%

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

   2. add-sqr-sqrt 53.8%

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

   3. sub-div 53.7%

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

9. Applied egg-rr 53.7%

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

10. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

12. Simplified 99.3%

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

    1. div-inv 99.2%

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

    2. associate-/l* 99.2%

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

    3. +-commutative 99.2%

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

14. Applied egg-rr 99.2%

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

    1. *-commutative 99.2%

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

    2. associate-/r/ 99.2%

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

    3. associate-*l/ 99.2%

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

    4. associate-*r* 99.2%

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

    5. times-frac 99.2%

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

    6. *-lft-identity 99.2%

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

    7. associate-/r* 99.4%

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

16. Simplified 99.4%

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

17. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Simplified 53.9%

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

   1. *-commutative 53.9%

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

   2. metadata-eval 53.9%

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

   3. distribute-lft-neg-in 53.9%

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

   4. distribute-rgt-neg-in 53.9%

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

   5. *-commutative 53.9%

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

   6. fma-neg 53.9%

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

   7. flip-- 53.8%

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

   8. div-sub 53.8%

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

   9. pow2 53.8%

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

   10. pow2 53.8%

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

   11. pow-prod-up 53.5%

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

   12. metadata-eval 53.5%

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

   13. fma-def 53.8%

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

   14. associate-*l* 53.8%

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

   15. pow2 53.8%

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

   16. associate-*l* 53.8%

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

   17. fma-def 53.8%

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

   18. associate-*l* 53.8%

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

5. Applied egg-rr 53.8%

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

   1. fma-def 53.5%

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

   2. +-commutative 53.5%

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

   3. *-commutative 53.5%

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

   4. fma-def 53.5%

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

   5. *-commutative 53.5%

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

   6. *-commutative 53.5%

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

   7. associate-*l* 53.5%

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

   8. fma-def 53.5%

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

   9. +-commutative 53.5%

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

   10. *-commutative 53.5%

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

   11. fma-def 53.5%

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

7. Simplified 53.5%

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

   1. flip-- 53.3%

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

   2. add-sqr-sqrt 53.8%

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

   3. sub-div 53.7%

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

9. Applied egg-rr 53.7%

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

10. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

12. Simplified 99.3%

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

13. Taylor expanded in b around 0 99.3%

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

    1. fma-def 99.3%

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

    2. unpow2 99.3%

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

15. Simplified 99.3%

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

16. Final simplification 99.3%

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

Alternative 3: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

3. Simplified 53.9%

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

   1. *-commutative 53.9%

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

   2. metadata-eval 53.9%

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

   3. distribute-lft-neg-in 53.9%

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

   4. distribute-rgt-neg-in 53.9%

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

   5. *-commutative 53.9%

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

   6. fma-neg 53.9%

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

   7. flip-- 53.8%

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

   8. div-sub 53.8%

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

   9. pow2 53.8%

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

   10. pow2 53.8%

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

   11. pow-prod-up 53.5%

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

   12. metadata-eval 53.5%

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

   13. fma-def 53.8%

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

   14. associate-*l* 53.8%

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

   15. pow2 53.8%

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

   16. associate-*l* 53.8%

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

   17. fma-def 53.8%

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

   18. associate-*l* 53.8%

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

5. Applied egg-rr 53.8%

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

   1. fma-def 53.5%

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

   2. +-commutative 53.5%

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

   3. *-commutative 53.5%

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

   4. fma-def 53.5%

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

   5. *-commutative 53.5%

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

   6. *-commutative 53.5%

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

   7. associate-*l* 53.5%

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

   8. fma-def 53.5%

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

   9. +-commutative 53.5%

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

   10. *-commutative 53.5%

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

   11. fma-def 53.5%

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

7. Simplified 53.5%

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

   1. flip-- 53.3%

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

   2. add-sqr-sqrt 53.8%

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

   3. sub-div 53.7%

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

9. Applied egg-rr 53.7%

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

10. Taylor expanded in b around 0 99.3%

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

    1. *-commutative 99.3%

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

12. Simplified 99.3%

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

13. Taylor expanded in b around 0 99.3%

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

    1. +-commutative 99.3%

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

    2. unpow2 99.3%

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

    3. fma-def 99.3%

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

    4. *-commutative 99.3%

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

    5. associate-*l* 99.3%

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

15. Simplified 99.3%

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

16. Final simplification 99.3%

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

Alternative 4: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 12:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b \cdot \left(b \cdot b\right)} \cdot \left(c \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 12.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a (* b (* b b))) (* c c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 12.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / (b * (b * b))) * (c * c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 12.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / Float64(b * Float64(b * b))) * Float64(c * c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 12.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 12

   1. Initial program 79.4%

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

   3. Simplified 79.4%

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

   if 12 < b

   1. Initial program 47.3%

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

   2. Taylor expanded in b around inf 86.7%

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

      1. mul-1-neg 86.7%

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

      2. unsub-neg 86.7%

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

      3. mul-1-neg 86.7%

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

      4. distribute-neg-frac 86.7%

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

      5. associate-/l* 86.7%

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

      6. associate-/r/ 86.7%

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

      7. unpow2 86.7%

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

   4. Simplified 86.7%

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

      1. unpow3 86.7%

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

   6. Applied egg-rr 86.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 12:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b \cdot \left(b \cdot b\right)} \cdot \left(c \cdot c\right)\\ \end{array} \]

Alternative 5: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 12:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b \cdot \left(b \cdot b\right)} \cdot \left(c \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 12.0)
   (/ (- (sqrt (- (* b b) (* (* a c) 4.0))) b) (* a 2.0))
   (- (/ (- c) b) (* (/ a (* b (* b b))) (* c c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 12.0) {
		tmp = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / (b * (b * b))) * (c * 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 <= 12.0d0) then
        tmp = (sqrt(((b * b) - ((a * c) * 4.0d0))) - b) / (a * 2.0d0)
    else
        tmp = (-c / b) - ((a / (b * (b * b))) * (c * c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 12.0) {
		tmp = (Math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / (b * (b * b))) * (c * c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 12.0:
		tmp = (math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0)
	else:
		tmp = (-c / b) - ((a / (b * (b * b))) * (c * c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 12.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / Float64(b * Float64(b * b))) * Float64(c * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 12.0)
		tmp = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	else
		tmp = (-c / b) - ((a / (b * (b * b))) * (c * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 12.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 12

   1. Initial program 79.4%

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

   3. Simplified 79.4%

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

      1. *-commutative 79.4%

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

      2. metadata-eval 79.4%

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

      3. distribute-lft-neg-in 79.4%

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

      4. distribute-rgt-neg-in 79.4%

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

      5. *-commutative 79.4%

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

      6. fma-neg 79.4%

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

      7. associate-*l* 79.4%

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

   5. Applied egg-rr 79.4%

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

   if 12 < b

   1. Initial program 47.3%

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

   2. Taylor expanded in b around inf 86.7%

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

      1. mul-1-neg 86.7%

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

      2. unsub-neg 86.7%

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

      3. mul-1-neg 86.7%

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

      4. distribute-neg-frac 86.7%

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

      5. associate-/l* 86.7%

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

      6. associate-/r/ 86.7%

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

      7. unpow2 86.7%

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

   4. Simplified 86.7%

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

      1. unpow3 86.7%

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

   6. Applied egg-rr 86.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 12:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b \cdot \left(b \cdot b\right)} \cdot \left(c \cdot c\right)\\ \end{array} \]

Alternative 6: 81.3% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

2. Taylor expanded in b around inf 81.4%

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

   1. mul-1-neg 81.4%

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

   2. unsub-neg 81.4%

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

   3. mul-1-neg 81.4%

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

   4. distribute-neg-frac 81.4%

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

   5. associate-/l* 81.4%

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

   6. associate-/r/ 81.4%

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

   7. unpow2 81.4%

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

4. Simplified 81.4%

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

   1. unpow3 81.4%

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

6. Applied egg-rr 81.4%

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

7. Final simplification 81.4%

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

Alternative 7: 64.1% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

2. Taylor expanded in b around inf 65.4%

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

   1. mul-1-neg 65.4%

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

   2. distribute-neg-frac 65.4%

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

4. Simplified 65.4%

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

5. Final simplification 65.4%

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

Alternative 8: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.9%

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

   1. add-sqr-sqrt 53.9%

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

   2. difference-of-squares 53.9%

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

   3. associate-*l* 53.9%

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

   4. sqrt-prod 53.9%

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

   5. metadata-eval 53.9%

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

   6. associate-*l* 53.9%

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

   7. sqrt-prod 53.9%

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

   8. metadata-eval 53.9%

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

3. Applied egg-rr 53.9%

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

   1. *-commutative 53.9%

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

   2. cancel-sign-sub-inv 53.9%

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

   3. metadata-eval 53.9%

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

5. Simplified 53.9%

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

6. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.25 \cdot \frac{-2 \cdot \sqrt{a \cdot c} + 2 \cdot \sqrt{a \cdot c}}{a}} \]
7. Step-by-step derivation

   1. associate-*r/ 3.2%

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

   2. distribute-rgt-out 3.2%

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

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{a \cdot c} \cdot \color{blue}{0}\right)}{a} \]

   4. mul0-rgt 3.2%

      \[\leadsto \frac{0.25 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

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

Reproduce

herbie shell --seed 2023285 
(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)))