Quadratic roots, wide range

Percentage Accurate: 17.9% → 99.4%

Time: 8.8s

Alternatives: 5

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: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(N[(-4.0 * a), $MachinePrecision] * c + 0.0), $MachinePrecision] / N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 19.8

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

4. Applied rewrites 19.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 20.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. associate--l+ N/A

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

   9. lower-fma.f64 N/A

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

   10. +-inverses N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 95.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in a around 0

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

   1. distribute-lft-out N/A

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

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. div-add-rev N/A

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

   6. associate-/l* N/A

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

   7. distribute-lft-out N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 94.7%

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

Alternative 3: 95.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. associate-*l/ N/A

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

   6. distribute-neg-frac N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

5. Applied rewrites 94.3%

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

6. Taylor expanded in b around -inf

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

8. Applied rewrites 94.4%

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

Alternative 4: 90.4% 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(Float64(-c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 89.3

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

5. Applied rewrites 89.3%

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

Alternative 5: 3.3% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 19.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 19.8

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

4. Applied rewrites 19.8%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. frac-add N/A

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

   8. lower-/.f64 N/A

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

6. Applied rewrites 20.9%

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

7. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-inverses N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{0}}{a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. lower-/.f64 N/A

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

   10. fp-cancel-sign-sub-inv N/A

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

   11. metadata-eval N/A

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

   12. +-inverses 3.3

       \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Applied rewrites 3.3%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

10. Taylor expanded in a around 0

    \[\leadsto 0 \]
11. Step-by-step derivation

12. Applied rewrites 3.3%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(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)))