Quadratic roots, wide range

Percentage Accurate: 18.0% → 97.7%

Time: 13.1s

Alternatives: 8

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: 18.0% 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: 97.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (/ (* (* c (* c (* c c))) (* 20.0 a)) (* t_0 (* b t_0)))
    (* -0.25 (* a a))
    (fma
     (* a a)
     (/ (* c (* c (* c -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);
	return fma((((c * (c * (c * c))) * (20.0 * a)) / (t_0 * (b * t_0))), (-0.25 * (a * a)), fma((a * a), ((c * (c * (c * -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))
	return fma(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(20.0 * a)) / Float64(t_0 * Float64(b * t_0))), Float64(-0.25 * Float64(a * a)), fma(Float64(a * a), Float64(Float64(c * Float64(c * Float64(c * -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]}, N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(20.0 * a), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.25 * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(c * N[(c * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $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 20.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. Simplified 97.0%

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

7. Applied egg-rr 97.1%

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

Alternative 2: 97.7% accurate, 0.3× speedup

Math

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

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) (* t_0 (* 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) / (t_0 * (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 * Float64(c * -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(t_0 * 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 * N[(c * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $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[(t$95$0 * N[(b * t$95$0), $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 20.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. Simplified 97.0%

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

7. Applied egg-rr 97.0%

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

8. Final simplification 97.0%

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

Alternative 3: 96.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

5. Simplified 95.9%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. cube-mult N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 95.9

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

8. Simplified 95.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

10. Applied egg-rr 96.2%

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

Alternative 4: 96.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

5. Simplified 95.9%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. cube-mult N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 95.9

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

8. Simplified 95.9%

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

9. Final simplification 95.9%

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

Alternative 5: 96.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.5%

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

3. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

5. Simplified 95.9%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. cube-mult N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 95.9

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

8. Simplified 95.9%

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

9. Taylor expanded in a around 0

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

    1. lower-*.f64 N/A

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 95.9

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

11. Simplified 95.9%

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

Alternative 6: 95.3% 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 20.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. Simplified 97.0%

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

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 94.4

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

8. Simplified 94.4%

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

Alternative 7: 95.3% 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 20.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 94.4

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

5. Simplified 94.4%

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

6. Final simplification 94.4%

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

Alternative 8: 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 20.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 88.6

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

5. Simplified 88.6%

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

Reproduce

herbie shell --seed 2024208 
(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)))