Quadratic roots, medium range

Percentage Accurate: 31.6% → 99.8%

Time: 13.3s

Alternatives: 7

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: 31.6% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 30.7

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

4. Applied egg-rr 30.7%

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

6. Applied egg-rr 31.4%

   \[\leadsto \color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

7. Taylor expanded in b around 0

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

   1. lower-*.f64 99.7

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

9. Simplified 99.7%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. +-commutative N/A

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

    4. lift-*.f64 N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-*.f64 99.7

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

11. Applied egg-rr 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 30.7

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

4. Applied egg-rr 30.7%

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

6. Applied egg-rr 31.4%

   \[\leadsto \color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

7. Taylor expanded in b around 0

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

   1. lower-*.f64 99.7

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

9. Simplified 99.7%

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

    1. *-commutative N/A

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

    2. lift-neg.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-fma.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift--.f64 N/A

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

    8. associate-/l* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-/.f64 99.4

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

11. Applied egg-rr 99.4%

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

Alternative 3: 91.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 30.7

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

4. Applied egg-rr 30.7%

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

6. Applied egg-rr 31.4%

   \[\leadsto \color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}} \]

7. Taylor expanded in b around 0

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

   1. lower-*.f64 99.7

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

9. Simplified 99.7%

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

10. Taylor expanded in c around 0

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

    1. distribute-lft-out-- N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 90.5

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

12. Simplified 90.5%

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

Alternative 4: 90.6% 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 30.7%

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

3. Taylor expanded in 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 90.1

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

5. Simplified 90.1%

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

6. Final simplification 90.1%

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

Alternative 5: 90.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
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 93.0%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 89.8

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

8. Simplified 89.8%

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

9. Taylor expanded in b around 0

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

    1. lower-/.f64 N/A

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

    2. distribute-lft-out N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 N/A

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

    9. cube-mult N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 N/A

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

    12. unpow2 N/A

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

    13. lower-*.f64 89.8

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

11. Simplified 89.8%

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

12. Final simplification 89.8%

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

Alternative 6: 90.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]
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 93.0%

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

6. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. cube-mult N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 89.9

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

8. Simplified 89.9%

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

9. Taylor expanded in b around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. mul-1-neg N/A

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

    4. mul-1-neg N/A

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

    5. distribute-neg-out N/A

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

    6. lower-neg.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-fma.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. cube-mult N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. unpow2 N/A

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

    14. lower-*.f64 89.7

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

11. Simplified 89.7%

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

12. Final simplification 89.7%

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

Alternative 7: 81.1% 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 30.7%

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

3. Taylor expanded in 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 81.3

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

5. Simplified 81.3%

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

Reproduce

herbie shell --seed 2024207 
(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)))