Quadratic roots, wide range

Percentage Accurate: 17.8% → 99.9%

Time: 12.2s

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: 17.8% 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: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 15.7

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

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

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 15.7

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

4. Applied rewrites 15.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

6. Applied rewrites 16.2%

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

7. Taylor expanded in b around 0

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

   1. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 95.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 15.7

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

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

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 15.7

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

4. Applied rewrites 15.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

6. Applied rewrites 16.2%

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

7. Taylor expanded in b around 0

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

   1. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

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. lower-fma.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 96.6

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

12. Applied rewrites 96.6%

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

13. Final simplification 96.6%

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

Alternative 3: 95.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lower--.f64 15.7

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

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

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

   16. lower-*.f64 N/A

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

   17. metadata-eval 15.7

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

4. Applied rewrites 15.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

6. Applied rewrites 16.2%

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

7. Taylor expanded in b around 0

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

   1. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in c around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 96.6

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

12. Applied rewrites 96.6%

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

13. Final simplification 96.6%

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

Alternative 4: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 96.5

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

5. Applied rewrites 96.5%

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

6. Final simplification 96.5%

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

Alternative 5: 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(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 15.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 92.0

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

5. Applied rewrites 92.0%

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

Reproduce

herbie shell --seed 2024222 
(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)))