Quadratic roots, wide range

Percentage Accurate: 17.9% → 97.5%

Time: 14.9s

Alternatives: 10

Speedup: 3.6×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.9% 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: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := c \cdot \left(c \cdot c\right)\\ \mathsf{fma}\left(\frac{\left(c \cdot t\_1\right) \cdot \left(20 \cdot a\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot t\_0\right)\right)}, -0.25 \cdot \left(a \cdot a\right), \mathsf{fma}\left(t\_1, \left(a \cdot a\right) \cdot \frac{-2}{\left(b \cdot b\right) \cdot t\_0}, \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* c (* c c))))
   (fma
    (/ (* (* c t_1) (* 20.0 a)) (* b (* (* b b) (* b t_0))))
    (* -0.25 (* a a))
    (fma
     t_1
     (* (* a a) (/ -2.0 (* (* b b) t_0)))
     (/ (fma (* c c) (/ a (* b b)) c) (- b))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c * (c * c);
	return fma((((c * t_1) * (20.0 * a)) / (b * ((b * b) * (b * t_0)))), (-0.25 * (a * a)), fma(t_1, ((a * a) * (-2.0 / ((b * b) * t_0))), (fma((c * c), (a / (b * b)), c) / -b)));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(c * Float64(c * c))
	return fma(Float64(Float64(Float64(c * t_1) * Float64(20.0 * a)) / Float64(b * Float64(Float64(b * b) * Float64(b * t_0)))), Float64(-0.25 * Float64(a * a)), fma(t_1, Float64(Float64(a * a) * Float64(-2.0 / Float64(Float64(b * b) * t_0))), Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(c * t$95$1), $MachinePrecision] * N[(20.0 * a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(N[(a * a), $MachinePrecision] * N[(-2.0 / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 96.8%

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

7. Applied rewrites 96.8%

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

8. Final simplification 96.8%

   \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(20 \cdot a\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, -0.25 \cdot \left(a \cdot a\right), \mathsf{fma}\left(c \cdot \left(c \cdot c\right), \left(a \cdot a\right) \cdot \frac{-2}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\right)\right) \]
9. Add Preprocessing

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-2}{\left(b \cdot b\right) \cdot t\_0}, \frac{\left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(20 \cdot a\right)\right) \cdot -0.25}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot t\_0\right)\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (fma
     c
     (* (* c c) (/ -2.0 (* (* b b) t_0)))
     (/
      (* (* (* c (* c (* c c))) (* 20.0 a)) -0.25)
      (* b (* (* b b) (* b t_0)))))
    (* a a)
    (/ (fma (* c c) (/ a (* b b)) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma(fma(c, ((c * c) * (-2.0 / ((b * b) * t_0))), ((((c * (c * (c * c))) * (20.0 * a)) * -0.25) / (b * ((b * b) * (b * t_0))))), (a * a), (fma((c * c), (a / (b * b)), c) / -b));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(fma(c, Float64(Float64(c * c) * Float64(-2.0 / Float64(Float64(b * b) * t_0))), Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(20.0 * a)) * -0.25) / Float64(b * Float64(Float64(b * b) * Float64(b * t_0))))), Float64(a * a), Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * N[(N[(c * c), $MachinePrecision] * N[(-2.0 / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(20.0 * a), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 96.8%

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

7. Applied rewrites 96.8%

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

8. Final simplification 96.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-2}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(20 \cdot a\right)\right) \cdot -0.25}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), a \cdot a, \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\right) \]
9. Add Preprocessing

Alternative 3: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathsf{fma}\left(\frac{-0.25 \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(20 \cdot a\right)\right)\right)}{b \cdot \left(t\_0 \cdot t\_0\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (/ (* -0.25 (* (* c (* c c)) (* c (* 20.0 a)))) (* b (* t_0 t_0)))
    (* a a)
    (/
     (- (/ (* (* c c) (fma -2.0 (/ (* c (* a a)) (* b b)) (- a))) (* b b)) c)
     b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma(((-0.25 * ((c * (c * c)) * (c * (20.0 * a)))) / (b * (t_0 * t_0))), (a * a), (((((c * c) * fma(-2.0, ((c * (a * a)) / (b * b)), -a)) / (b * b)) - c) / b));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(Float64(Float64(-0.25 * Float64(Float64(c * Float64(c * c)) * Float64(c * Float64(20.0 * a)))) / Float64(b * Float64(t_0 * t_0))), Float64(a * a), Float64(Float64(Float64(Float64(Float64(c * c) * fma(-2.0, Float64(Float64(c * Float64(a * a)) / Float64(b * b)), Float64(-a))) / Float64(b * b)) - c) / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(-0.25 * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(c * N[(20.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(-2.0 * N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 96.8%

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

7. Applied rewrites 96.8%

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

9. Applied rewrites 96.8%

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

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\color{blue}{{c}^{2} \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} - a\right)}}{b \cdot b} - c}{b}\right) \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\color{blue}{{c}^{2} \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} - a\right)}}{b \cdot b} - c}{b}\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\color{blue}{\left(c \cdot c\right)} \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} - a\right)}{b \cdot b} - c}{b}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\color{blue}{\left(c \cdot c\right)} \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} - a\right)}{b \cdot b} - c}{b}\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \color{blue}{\left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(a\right)\right)\right)}}{b \cdot b} - c}{b}\right) \]

    5. neg-mul-1 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \color{blue}{-1 \cdot a}\right)}{b \cdot b} - c}{b}\right) \]

    6. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot c}{{b}^{2}}, -1 \cdot a\right)}}{b \cdot b} - c}{b}\right) \]

    7. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{a}^{2} \cdot c}{{b}^{2}}}, -1 \cdot a\right)}{b \cdot b} - c}{b}\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{c \cdot {a}^{2}}}{{b}^{2}}, -1 \cdot a\right)}{b \cdot b} - c}{b}\right) \]

    9. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{\color{blue}{c \cdot {a}^{2}}}{{b}^{2}}, -1 \cdot a\right)}{b \cdot b} - c}{b}\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \color{blue}{\left(a \cdot a\right)}}{{b}^{2}}, -1 \cdot a\right)}{b \cdot b} - c}{b}\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \color{blue}{\left(a \cdot a\right)}}{{b}^{2}}, -1 \cdot a\right)}{b \cdot b} - c}{b}\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{\color{blue}{b \cdot b}}, -1 \cdot a\right)}{b \cdot b} - c}{b}\right) \]

    13. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{\color{blue}{b \cdot b}}, -1 \cdot a\right)}{b \cdot b} - c}{b}\right) \]

    14. neg-mul-1 N/A

        \[\leadsto \mathsf{fma}\left(\frac{\left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 20\right)\right)\right) \cdot \frac{-1}{4}}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, \color{blue}{\mathsf{neg}\left(a\right)}\right)}{b \cdot b} - c}{b}\right) \]

    15. lower-neg.f64 96.8

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

12. Applied rewrites 96.8%

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

13. Final simplification 96.8%

    \[\leadsto \mathsf{fma}\left(\frac{-0.25 \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(20 \cdot a\right)\right)\right)}{b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}, a \cdot a, \frac{\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, -a\right)}{b \cdot b} - c}{b}\right) \]
14. Add Preprocessing

Alternative 4: 94.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -5:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(a \cdot 2\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -5

   1. Initial program 77.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   7. Applied rewrites 78.5%

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

   if -5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 12.7%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 97.3

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

   8. Applied rewrites 97.3%

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

4. Final simplification 95.3%

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

Alternative 5: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.3%

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

9. Applied rewrites 95.6%

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

10. Final simplification 95.6%

    \[\leadsto \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{\left(a \cdot a\right) \cdot -2}{b \cdot b} - c \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot b\right)} - \frac{c}{b} \]
11. Add Preprocessing

Alternative 6: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

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

8. Final simplification 95.5%

   \[\leadsto \frac{\frac{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)}{b \cdot b} - c \cdot \left(c \cdot a\right)}{b \cdot b} - c}{b} \]
9. Add Preprocessing

Alternative 7: 96.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. sub-neg N/A

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

   5. neg-mul-1 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. neg-mul-1 N/A

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

   15. lower-neg.f64 95.3

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

10. Applied rewrites 95.3%

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

11. Final simplification 95.3%

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

Alternative 8: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 96.8%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

   4. lower-neg.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 93.8

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

8. Applied rewrites 93.8%

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

Alternative 9: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-/l* N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 93.7

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

5. Applied rewrites 93.7%

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

6. Final simplification 93.7%

   \[\leadsto \frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b} \]
7. Add Preprocessing

Alternative 10: 90.3% accurate, 3.6× 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 19.5%

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

3. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{c}{b}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{c}{\mathsf{neg}\left(b\right)}} \]

   4. lower-neg.f64 89.0

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

5. Applied rewrites 89.0%

   \[\leadsto \color{blue}{\frac{c}{-b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))