Quadratic roots, medium range

Percentage Accurate: 31.9% → 99.6%

Time: 11.6s

Alternatives: 8

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: 31.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(0.5, a \cdot \left(c \cdot -4\right), 0\right)}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (fma 0.5 (* a (* c -4.0)) 0.0) a)
  (+ (sqrt (fma (* c -4.0) a (* b b))) b)))

C

double code(double a, double b, double c) {
	return (fma(0.5, (a * (c * -4.0)), 0.0) / a) / (sqrt(fma((c * -4.0), a, (b * b))) + b);
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(0.5, Float64(a * Float64(c * -4.0)), 0.0) / a) / Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) + b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(0.5 * N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] / a), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.7%

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

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 28.7

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

4. Applied rewrites 28.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. sub-neg N/A

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

   12. flip-- N/A

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

   13. lift-+.f64 N/A

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(0.5, \left(-4 \cdot c\right) \cdot a, 0\right)}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}} \]

7. Final simplification 99.7%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(0.5, a \cdot \left(c \cdot -4\right), 0\right)}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b} \cdot \frac{\mathsf{fma}\left(c \cdot -4, a, 0\right)}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ 0.5 (+ (sqrt (fma (* c -4.0) a (* b b))) b))
  (/ (fma (* c -4.0) a 0.0) a)))

C

double code(double a, double b, double c) {
	return (0.5 / (sqrt(fma((c * -4.0), a, (b * b))) + b)) * (fma((c * -4.0), a, 0.0) / a);
}

Julia

function code(a, b, c)
	return Float64(Float64(0.5 / Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) + b)) * Float64(fma(Float64(c * -4.0), a, 0.0) / a))
end

Wolfram

code[a_, b_, c_] := N[(N[(0.5 / N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * -4.0), $MachinePrecision] * a + 0.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.7%

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

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 28.7

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

4. Applied rewrites 28.7%

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

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{a} \cdot \frac{0.5}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}} \]

7. Final simplification 99.5%

   \[\leadsto \frac{0.5}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b} \cdot \frac{\mathsf{fma}\left(c \cdot -4, a, 0\right)}{a} \]
8. Add Preprocessing

Alternative 3: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 28.7

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 28.7

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

4. Applied rewrites 28.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. frac-times N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 29.4%

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

7. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.4

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 90.4% accurate, 1.1× speedup

Math

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

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(Float64(b * b) * b)), Float64(c / b)))
end

Wolfram

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

Derivation

1. Initial program 28.7%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 95.6%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-out N/A

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

   5. lower-neg.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow3 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 92.2

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

8. Applied rewrites 92.2%

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

Alternative 5: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 28.7

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-neg.f64 N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 28.7

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

4. Applied rewrites 28.7%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 92.2

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

7. Applied rewrites 92.2%

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

8. Final simplification 92.2%

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

Alternative 6: 90.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 92.1

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

5. Applied rewrites 92.1%

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

6. Final simplification 92.1%

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

Alternative 7: 90.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 94.1%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 91.9%

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

Alternative 8: 80.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.7%

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

3. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 83.7

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

5. Applied rewrites 83.7%

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

Reproduce

herbie shell --seed 2024235 
(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)))