Quadratic roots, medium range

Percentage Accurate: 31.1% → 95.4%

Time: 14.6s

Alternatives: 6

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.1% 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: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot c\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(t\_0, -2 \cdot {b}^{-5}, \frac{\left(c \cdot t\_0\right) \cdot \left(a \cdot -5\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \frac{c \cdot c}{\left(b \cdot b\right) \cdot \left(-b\right)}\right), a, \frac{c}{-b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* c c))))
   (fma
    (fma
     a
     (fma
      t_0
      (* -2.0 (pow b -5.0))
      (/ (* (* c t_0) (* a -5.0)) (* b (* (* b b) (* b (* b (* b b)))))))
     (/ (* c c) (* (* b b) (- b))))
    a
    (/ c (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = c * (c * c);
	return fma(fma(a, fma(t_0, (-2.0 * pow(b, -5.0)), (((c * t_0) * (a * -5.0)) / (b * ((b * b) * (b * (b * (b * b))))))), ((c * c) / ((b * b) * -b))), a, (c / -b));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(c * c))
	return fma(fma(a, fma(t_0, Float64(-2.0 * (b ^ -5.0)), Float64(Float64(Float64(c * t_0) * Float64(a * -5.0)) / Float64(b * Float64(Float64(b * b) * Float64(b * Float64(b * Float64(b * b))))))), Float64(Float64(c * c) / Float64(Float64(b * b) * Float64(-b)))), a, Float64(c / Float64(-b)))
end

Wolfram

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

Derivation

1. Initial program 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 96.3%

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

6. Taylor expanded in a around 0

   \[\leadsto a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{7}} + -2 \cdot \frac{{c}^{3}}{{b}^{5}}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \color{blue}{\frac{c}{b}} \]
7. Step-by-step derivation

8. Applied rewrites 96.4%

   \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{a \cdot {c}^{4}}{{b}^{7}}, -5, \frac{-2 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{5}}\right), -\frac{c \cdot c}{b \cdot \left(b \cdot b\right)}\right)}, \frac{-c}{b}\right) \]
9. Step-by-step derivation

10. Applied rewrites 96.4%

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

12. Applied rewrites 96.4%

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

13. Final simplification 96.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(c \cdot \left(c \cdot c\right), -2 \cdot {b}^{-5}, \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -5\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\right), \frac{c \cdot c}{\left(b \cdot b\right) \cdot \left(-b\right)}\right), a, \frac{c}{-b}\right) \]
14. Add Preprocessing

Alternative 2: 95.4% accurate, 0.3× speedup

Math

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

FPCore

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

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * Float64(b * Float64(b * b)))
	return fma(fma(c, Float64(Float64(c * c) * Float64(-2.0 / t_0)), Float64(-0.25 * Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * 20.0)) / Float64(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[(N[(b * b), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * N[(N[(c * c), $MachinePrecision] * N[(-2.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 20.0), $MachinePrecision]), $MachinePrecision] / N[(b * 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 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 96.3%

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

7. Applied rewrites 96.3%

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

8. Final simplification 96.3%

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

Alternative 3: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{1}{b}\right), \frac{b}{-c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma a (fma (* a -2.0) (* (/ c (* b (* b b))) -0.5) (/ 1.0 b)) (/ b (- c)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(a, fma((a * -2.0), ((c / (b * (b * b))) * -0.5), (1.0 / b)), (b / -c));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(a, fma(Float64(a * -2.0), Float64(Float64(c / Float64(b * Float64(b * b))) * -0.5), Float64(1.0 / b)), Float64(b / Float64(-c))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(a * N[(N[(a * -2.0), $MachinePrecision] * N[(N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[(b / (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 30.8

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

4. Applied rewrites 30.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 30.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 30.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-neg.f64 N/A

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

   13. unsub-neg N/A

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

   14. lower--.f64 30.8

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

6. Applied rewrites 30.8%

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

7. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{b}{c}\right)}} \]

   2. lower-neg.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{b}{c}\right)}} \]

   3. lower-/.f64 82.0

      \[\leadsto \frac{1}{-\color{blue}{\frac{b}{c}}} \]

9. Applied rewrites 82.0%

   \[\leadsto \frac{1}{\color{blue}{-\frac{b}{c}}} \]

10. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. associate-*r* N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. distribute-rgt-out N/A

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

    7. metadata-eval N/A

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

    8. lower-*.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. cube-mult N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. unpow2 N/A

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

    14. lower-*.f64 N/A

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

    15. lower-/.f64 N/A

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

12. Applied rewrites 94.8%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2 \cdot a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{1}{b}\right), \frac{-b}{c}\right)}} \]

13. Final simplification 94.8%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -2, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.5, \frac{1}{b}\right), \frac{b}{-c}\right)} \]
14. Add Preprocessing

Alternative 4: 93.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return Float64(c * fma(c, Float64(fma(-2.0, Float64(Float64(c * Float64(a * a)) / 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[(-2.0 * N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + (-a)), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. Applied rewrites 94.5%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 94.5%

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

Alternative 5: 90.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / ((a / b) - (b / c))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / c));
}

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((a / b) - (b / c));
end

Wolfram

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

Derivation

1. Initial program 30.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 30.8

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

4. Applied rewrites 30.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 30.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 30.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-neg.f64 N/A

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

   13. unsub-neg N/A

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

   14. lower--.f64 30.8

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

6. Applied rewrites 30.8%

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

7. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 91.8

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

9. Applied rewrites 91.8%

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

10. Final simplification 91.8%

    \[\leadsto \frac{1}{\frac{a}{b} - \frac{b}{c}} \]
11. Add Preprocessing

Alternative 6: 81.5% 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 30.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 82.2

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

5. Applied rewrites 82.2%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))