Quadratic roots, medium range

Percentage Accurate: 31.7% → 99.4%

Time: 13.1s

Alternatives: 5

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: 31.7% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.3%

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

4. Applied egg-rr 28.3%

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

   1. *-commutative N/A

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

   2. flip-- N/A

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

   3. frac-times N/A

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

   4. /-lowering-/.f64 N/A

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

6. Applied egg-rr 29.2%

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

7. Taylor expanded in b around 0

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. *-lowering-*.f64 99.4

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

9. Simplified 99.4%

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

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

    3. *-lowering-*.f64 N/A

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.3%

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

4. Applied egg-rr 28.3%

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

   1. *-commutative N/A

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

   2. flip-- N/A

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

   3. frac-times N/A

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

   4. /-lowering-/.f64 N/A

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

6. Applied egg-rr 29.2%

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

7. Taylor expanded in b around 0

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

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

   3. *-lowering-*.f64 99.4

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

9. Simplified 99.4%

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

    1. *-commutative N/A

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

    2. associate-/l* N/A

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

    3. *-lowering-*.f64 N/A

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

    4. /-lowering-/.f64 N/A

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

    5. *-lowering-*.f64 N/A

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

    6. *-commutative N/A

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

    7. *-lowering-*.f64 N/A

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

    8. +-lowering-+.f64 N/A

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

    9. sqrt-lowering-sqrt.f64 N/A

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

    10. associate-*r* N/A

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

    11. *-commutative N/A

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

    12. associate-*l* N/A

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

    13. accelerator-lowering-fma.f64 N/A

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

    14. *-lowering-*.f64 N/A

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

    15. *-lowering-*.f64 99.4

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.3%

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

3. Taylor expanded in a around 0

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

5. Simplified 96.8%

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

6. Taylor expanded in a around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

   4. +-commutative N/A

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

   5. neg-lowering-neg.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 N/A

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

   17. /-lowering-/.f64 92.5

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

8. Simplified 92.5%

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

Alternative 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 28.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. neg-lowering-neg.f64 N/A

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

   5. /-lowering-/.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. accelerator-lowering-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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.f64 92.5

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

5. Simplified 92.5%

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

6. Final simplification 92.5%

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

Alternative 5: 81.0% 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 28.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. /-lowering-/.f64 N/A

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

   4. neg-lowering-neg.f64 83.9

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

5. Simplified 83.9%

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

Reproduce

herbie shell --seed 2024199 
(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)))