Quadratic roots, medium range

Percentage Accurate: 30.9% → 99.3%

Time: 16.3s

Alternatives: 5

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 30.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.9%

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

   1. *-commutative 32.9%

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

   2. +-commutative 32.9%

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

   3. sqr-neg 32.9%

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

   4. unsub-neg 32.9%

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

   5. sqr-neg 32.9%

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

   6. fma-neg 33.0%

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

   7. distribute-lft-neg-in 33.0%

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

   8. *-commutative 33.0%

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

   9. *-commutative 33.0%

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

   10. distribute-rgt-neg-in 33.0%

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

   11. metadata-eval 33.0%

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

3. Simplified 33.0%

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

   1. *-commutative 33.0%

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

   2. metadata-eval 33.0%

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

   3. distribute-lft-neg-in 33.0%

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

   4. distribute-rgt-neg-in 33.0%

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

   5. *-commutative 33.0%

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

   6. fma-neg 32.9%

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

   7. flip-- 32.8%

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

   8. div-sub 32.8%

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

   9. pow2 32.8%

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

   10. pow2 32.8%

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

   11. pow-prod-up 32.8%

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

   12. metadata-eval 32.8%

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

   13. fma-define 32.9%

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

   14. associate-*l* 32.9%

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

   15. pow2 32.9%

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

   16. associate-*l* 32.9%

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

   17. fma-define 32.9%

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

   18. associate-*l* 32.9%

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

6. Applied egg-rr 32.9%

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

   1. flip-- 32.8%

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

8. Applied egg-rr 33.9%

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

   1. metadata-eval 33.9%

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

   2. unpow2 33.9%

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

   3. swap-sqr 33.9%

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

   4. unpow2 33.9%

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

   5. associate-*r* 33.9%

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

   6. *-commutative 33.9%

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

   7. associate-*l* 33.9%

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

   8. associate-*r* 33.9%

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

   9. *-commutative 33.9%

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

   10. associate-*l* 33.9%

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

   11. +-commutative 33.9%

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

10. Simplified 33.9%

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

11. Taylor expanded in a around inf 99.3%

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

    1. associate-*r* 99.3%

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

    2. fma-define 99.3%

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

    3. distribute-lft1-in 99.3%

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

    4. metadata-eval 99.3%

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

    5. mul0-lft 99.3%

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

    6. metadata-eval 99.3%

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

13. Simplified 99.3%

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

14. Final simplification 99.3%

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

Alternative 2: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) - ((c / b) + ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) - ((c / b) + ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) - ((c / b) + ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - ((c / b) + ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

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

Derivation

1. Initial program 32.9%

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

   1. *-commutative 32.9%

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

3. Simplified 32.9%

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

5. Taylor expanded in b around inf 93.3%

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

6. Final simplification 93.3%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]
7. Add Preprocessing

Alternative 3: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -0.0002:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -0.0002) t_0 (/ c (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0002) {
		tmp = t_0;
	} else {
		tmp = 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 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.0002d0)) then
        tmp = t_0
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.0002) {
		tmp = t_0;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.0002:
		tmp = t_0
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.0002)
		tmp = t_0;
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.0002)
		tmp = t_0;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -0.0002], t$95$0, N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -2.0000000000000001e-4

   1. Initial program 71.4%

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

   if -2.0000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 20.3%

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

      1. *-commutative 20.3%

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

   3. Simplified 20.3%

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

   5. Taylor expanded in b around inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. distribute-neg-frac2 89.3%

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

   7. Simplified 89.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -0.0002:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} - {c}^{2} \cdot \frac{a}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (- (/ c (- b)) (* (pow c 2.0) (/ a (pow b 3.0)))))

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c / -b) - ((c ** 2.0d0) * (a / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (c / -b) - (Math.pow(c, 2.0) * (a / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (c / -b) - (math.pow(c, 2.0) * (a / math.pow(b, 3.0)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (c / -b) - ((c ^ 2.0) * (a / (b ^ 3.0)));
end

Wolfram

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

Derivation

1. Initial program 32.9%

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

   1. *-commutative 32.9%

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

3. Simplified 32.9%

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

5. Taylor expanded in b around inf 89.3%

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

   1. distribute-lft-out 89.3%

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

   2. associate-/l* 89.5%

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

   3. associate-/l* 89.5%

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

7. Simplified 89.5%

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

8. Taylor expanded in a around 0 89.6%

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

   1. mul-1-neg 89.6%

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

   2. unsub-neg 89.6%

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

   3. mul-1-neg 89.6%

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

   4. distribute-neg-frac2 89.6%

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

   5. *-commutative 89.6%

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

   6. associate-/l* 89.6%

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

10. Simplified 89.6%

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

11. Final simplification 89.6%

    \[\leadsto \frac{c}{-b} - {c}^{2} \cdot \frac{a}{{b}^{3}} \]
12. Add Preprocessing

Alternative 5: 81.7% 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(c / Float64(-b))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.9%

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

   1. *-commutative 32.9%

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

3. Simplified 32.9%

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

5. Taylor expanded in b around inf 79.9%

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

   1. mul-1-neg 79.9%

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

   2. distribute-neg-frac2 79.9%

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

7. Simplified 79.9%

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

8. Final simplification 79.9%

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

Reproduce

herbie shell --seed 2024046 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))