Quadratic roots, narrow range

Percentage Accurate: 55.0% → 99.4%

Time: 14.5s

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: 55.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Simplified 53.6%

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

   1. *-commutative 53.6%

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

   2. metadata-eval 53.6%

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

   3. distribute-lft-neg-in 53.6%

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

   4. distribute-rgt-neg-in 53.6%

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

   5. *-commutative 53.6%

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

   6. fma-neg 53.5%

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

   7. flip-- 53.3%

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

   8. div-sub 53.2%

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

   9. pow2 53.2%

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

   10. pow2 53.2%

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

   11. pow-prod-up 53.4%

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

   12. metadata-eval 53.4%

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

   13. fma-def 53.5%

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

   14. associate-*l* 53.5%

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

   15. pow2 53.5%

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

   16. associate-*l* 53.5%

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

   17. fma-def 53.6%

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

   18. associate-*l* 53.6%

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

5. Applied egg-rr 53.6%

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

   1. fma-def 53.4%

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

   2. +-commutative 53.4%

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

   3. *-commutative 53.4%

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

   4. fma-def 53.4%

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

   5. *-commutative 53.4%

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

   6. *-commutative 53.4%

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

   7. associate-*l* 53.4%

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

   8. fma-def 53.4%

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

   9. +-commutative 53.4%

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

   10. *-commutative 53.4%

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

   11. fma-def 53.4%

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

7. Simplified 53.4%

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

   1. flip-- 53.0%

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

   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 53.8%

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

9. Applied egg-rr 53.8%

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

11. Simplified 53.8%

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

12. Taylor expanded in b around 0 99.3%

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

    1. associate-*r* 99.3%

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

14. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. associate-/l/ 99.3%

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

    3. associate-*l* 99.3%

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

    4. cancel-sign-sub-inv 99.3%

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

    5. metadata-eval 99.3%

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

16. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{1 \cdot \frac{-4 \cdot \left(a \cdot c\right)}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{\frac{{b}^{4} + -16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right)}}\right)}} \]
17. Step-by-step derivation

    1. *-lft-identity 99.3%

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

    2. associate-/r* 99.4%

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

    3. *-commutative 99.4%

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

18. Simplified 99.4%

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

19. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Simplified 53.6%

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

   1. *-commutative 53.6%

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

   2. metadata-eval 53.6%

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

   3. distribute-lft-neg-in 53.6%

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

   4. distribute-rgt-neg-in 53.6%

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

   5. *-commutative 53.6%

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

   6. fma-neg 53.5%

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

   7. flip-- 53.3%

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

   8. div-sub 53.2%

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

   9. pow2 53.2%

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

   10. pow2 53.2%

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

   11. pow-prod-up 53.4%

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

   12. metadata-eval 53.4%

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

   13. fma-def 53.5%

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

   14. associate-*l* 53.5%

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

   15. pow2 53.5%

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

   16. associate-*l* 53.5%

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

   17. fma-def 53.6%

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

   18. associate-*l* 53.6%

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

5. Applied egg-rr 53.6%

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

   1. fma-def 53.4%

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

   2. +-commutative 53.4%

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

   3. *-commutative 53.4%

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

   4. fma-def 53.4%

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

   5. *-commutative 53.4%

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

   6. *-commutative 53.4%

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

   7. associate-*l* 53.4%

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

   8. fma-def 53.4%

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

   9. +-commutative 53.4%

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

   10. *-commutative 53.4%

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

   11. fma-def 53.4%

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

7. Simplified 53.4%

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

   1. flip-- 53.0%

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

   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 53.8%

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

9. Applied egg-rr 53.8%

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

11. Simplified 53.8%

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

12. Taylor expanded in b around 0 99.3%

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

    1. associate-*r* 99.3%

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

14. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. associate-/l* 99.3%

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

    3. cancel-sign-sub-inv 99.3%

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

    4. metadata-eval 99.3%

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

16. Applied egg-rr 99.3%

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{-4 \cdot a}{\frac{b + \sqrt{\frac{{b}^{4} + -16 \cdot {\left(a \cdot c\right)}^{2}}{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right)}}}{c}}}}{a \cdot 2} \]
17. Step-by-step derivation

    1. *-lft-identity 99.3%

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

    2. associate-/r/ 99.4%

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

    3. *-commutative 99.4%

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

18. Simplified 99.4%

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

19. Final simplification 99.4%

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

Alternative 3: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b + \left(a \cdot c\right) \cdot -4\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.007:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{b}^{3}} \cdot \left(-c \cdot c\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* b b) (* (* a c) -4.0))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.007)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* a 2.0))
     (- (* (/ a (pow b 3.0)) (- (* c c))) (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) + ((a * c) * -4.0);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.007) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((a / pow(b, 3.0)) * -(c * c)) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) + ((a * c) * (-4.0d0))
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-0.007d0)) then
        tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (a * 2.0d0)
    else
        tmp = ((a / (b ** 3.0d0)) * -(c * c)) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * b) + ((a * c) * -4.0);
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.007) {
		tmp = ((t_0 - (b * b)) / (b + Math.sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = ((a / Math.pow(b, 3.0)) * -(c * c)) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) + ((a * c) * -4.0)
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.007:
		tmp = ((t_0 - (b * b)) / (b + math.sqrt(t_0))) / (a * 2.0)
	else:
		tmp = ((a / math.pow(b, 3.0)) * -(c * c)) - (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.007)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a / (b ^ 3.0)) * Float64(-Float64(c * c))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) + ((a * c) * -4.0);
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.007)
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (a * 2.0);
	else
		tmp = ((a / (b ^ 3.0)) * -(c * c)) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.007], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * (-N[(c * c), $MachinePrecision])), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.00700000000000000015

   1. Initial program 76.8%

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

   3. Simplified 76.8%

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

      1. *-commutative 76.8%

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

      2. metadata-eval 76.8%

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

      3. distribute-lft-neg-in 76.8%

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

      4. distribute-rgt-neg-in 76.8%

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

      5. *-commutative 76.8%

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

      6. fma-neg 76.8%

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

      7. associate-*l* 76.8%

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

   5. Applied egg-rr 76.8%

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

      1. flip-- 76.8%

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

      2. add-sqr-sqrt 79.0%

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

      3. cancel-sign-sub-inv 79.0%

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

      4. metadata-eval 79.0%

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

      5. cancel-sign-sub-inv 79.0%

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

      6. metadata-eval 79.0%

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

   7. Applied egg-rr 79.0%

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

   if -0.00700000000000000015 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 44.9%

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

   2. Taylor expanded in b around inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

      3. mul-1-neg 88.8%

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

      4. distribute-neg-frac 88.8%

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

      5. associate-/l* 88.8%

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

      6. associate-/r/ 88.8%

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

      7. unpow2 88.8%

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

   4. Simplified 88.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.007:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + \left(a \cdot c\right) \cdot -4\right) - b \cdot b}{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{b}^{3}} \cdot \left(-c \cdot c\right) - \frac{c}{b}\\ \end{array} \]

Alternative 4: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.007:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{b}^{3}} \cdot \left(-c \cdot c\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -0.007)
   (/ (- (sqrt (- (* b b) (* (* a c) 4.0))) b) (* a 2.0))
   (- (* (/ a (pow b 3.0)) (- (* c c))) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.007) {
		tmp = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = ((a / pow(b, 3.0)) * -(c * c)) - (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)) <= (-0.007d0)) then
        tmp = (sqrt(((b * b) - ((a * c) * 4.0d0))) - b) / (a * 2.0d0)
    else
        tmp = ((a / (b ** 3.0d0)) * -(c * c)) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.007) {
		tmp = (Math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	} else {
		tmp = ((a / Math.pow(b, 3.0)) * -(c * c)) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.007:
		tmp = (math.sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0)
	else:
		tmp = ((a / math.pow(b, 3.0)) * -(c * c)) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -0.007)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * c) * 4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a / (b ^ 3.0)) * Float64(-Float64(c * c))) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -0.007)
		tmp = (sqrt(((b * b) - ((a * c) * 4.0))) - b) / (a * 2.0);
	else
		tmp = ((a / (b ^ 3.0)) * -(c * c)) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.007], 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[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * (-N[(c * c), $MachinePrecision])), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.00700000000000000015

   1. Initial program 76.8%

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

   3. Simplified 76.8%

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

      1. *-commutative 76.8%

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

      2. metadata-eval 76.8%

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

      3. distribute-lft-neg-in 76.8%

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

      4. distribute-rgt-neg-in 76.8%

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

      5. *-commutative 76.8%

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

      6. fma-neg 76.8%

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

      7. associate-*l* 76.8%

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

   5. Applied egg-rr 76.8%

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

   if -0.00700000000000000015 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 44.9%

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

   2. Taylor expanded in b around inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

      3. mul-1-neg 88.8%

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

      4. distribute-neg-frac 88.8%

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

      5. associate-/l* 88.8%

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

      6. associate-/r/ 88.8%

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

      7. unpow2 88.8%

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

   4. Simplified 88.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.007:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{b}^{3}} \cdot \left(-c \cdot c\right) - \frac{c}{b}\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(a * -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[(c * N[(a * -4.0), $MachinePrecision]), $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 53.5%

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

3. Simplified 53.6%

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

   1. *-commutative 53.6%

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

   2. metadata-eval 53.6%

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

   3. distribute-lft-neg-in 53.6%

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

   4. distribute-rgt-neg-in 53.6%

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

   5. *-commutative 53.6%

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

   6. fma-neg 53.5%

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

   7. flip-- 53.3%

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

   8. div-sub 53.2%

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

   9. pow2 53.2%

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

   10. pow2 53.2%

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

   11. pow-prod-up 53.4%

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

   12. metadata-eval 53.4%

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

   13. fma-def 53.5%

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

   14. associate-*l* 53.5%

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

   15. pow2 53.5%

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

   16. associate-*l* 53.5%

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

   17. fma-def 53.6%

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

   18. associate-*l* 53.6%

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

5. Applied egg-rr 53.6%

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

   1. fma-def 53.4%

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

   2. +-commutative 53.4%

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

   3. *-commutative 53.4%

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

   4. fma-def 53.4%

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

   5. *-commutative 53.4%

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

   6. *-commutative 53.4%

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

   7. associate-*l* 53.4%

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

   8. fma-def 53.4%

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

   9. +-commutative 53.4%

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

   10. *-commutative 53.4%

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

   11. fma-def 53.4%

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

7. Simplified 53.4%

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

   1. flip-- 53.0%

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

   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 53.8%

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

9. Applied egg-rr 53.8%

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

11. Simplified 53.8%

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

12. Taylor expanded in b around 0 99.3%

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

    1. associate-*r* 99.3%

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

14. Simplified 99.3%

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

15. Taylor expanded in b around 0 99.3%

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

    1. fma-def 99.3%

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

    2. *-commutative 99.3%

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

    3. unpow2 99.3%

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

17. Simplified 99.3%

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

18. Final simplification 99.3%

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

Alternative 6: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

2. Taylor expanded in b around inf 82.6%

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

   1. mul-1-neg 82.6%

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

   2. unsub-neg 82.6%

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

   3. mul-1-neg 82.6%

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

   4. distribute-neg-frac 82.6%

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

   5. associate-/l* 82.6%

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

   6. associate-/r/ 82.6%

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

   7. unpow2 82.6%

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

4. Simplified 82.6%

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

5. Final simplification 82.6%

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

Alternative 7: 64.7% 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 53.5%

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

2. Taylor expanded in b around inf 65.8%

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

   1. mul-1-neg 65.8%

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

   2. distribute-neg-frac 65.8%

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

4. Simplified 65.8%

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

5. Final simplification 65.8%

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

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 53.5%

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

   1. add-sqr-sqrt 53.5%

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

   2. difference-of-squares 53.6%

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

   3. associate-*l* 53.6%

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

   4. sqrt-prod 53.6%

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

   5. metadata-eval 53.6%

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

   6. associate-*l* 53.6%

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

   7. sqrt-prod 53.6%

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

   8. metadata-eval 53.6%

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

3. Applied egg-rr 53.6%

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

   1. *-commutative 53.6%

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

   2. cancel-sign-sub-inv 53.6%

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

   3. metadata-eval 53.6%

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

5. Simplified 53.6%

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

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 2023278 
(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)))