Quadratic roots, narrow range

Percentage Accurate: 55.3% → 99.5%

Time: 13.5s

Alternatives: 8

Speedup: 3.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.3% 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: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 57.8

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

4. Applied egg-rr 57.8%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

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

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

   4. rem-square-sqrt N/A

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

   5. sqr-neg N/A

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

   6. --lowering--.f64 N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 59.2%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

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

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

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

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

   5. associate--l+ N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. +-inverses N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. *-commutative N/A

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

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

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

   14. sub-neg N/A

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

   15. remove-double-neg N/A

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

   16. +-commutative N/A

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

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

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

8. Applied egg-rr 99.5%

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

Alternative 2: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 57.8

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

4. Applied egg-rr 57.8%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

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

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

   4. rem-square-sqrt N/A

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

   5. sqr-neg N/A

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

   6. --lowering--.f64 N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 59.2%

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

   1. associate-/l/ N/A

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

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

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. +-inverses N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

8. Applied egg-rr 99.3%

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

9. Final simplification 99.3%

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

Alternative 3: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 26.5:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 26.5)
   (* (/ -0.5 a) (- b (sqrt (fma b b (* c (* a -4.0))))))
   (- (fma a (/ (* c c) (* b (* b b))) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 26.5) {
		tmp = (-0.5 / a) * (b - sqrt(fma(b, b, (c * (a * -4.0)))));
	} else {
		tmp = -fma(a, ((c * c) / (b * (b * b))), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 26.5)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(fma(b, b, Float64(c * Float64(a * -4.0))))));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / Float64(b * Float64(b * b))), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 26.5], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 26.5

   1. Initial program 82.4%

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

   4. Applied egg-rr 82.6%

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

   if 26.5 < b

   1. Initial program 47.2%

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

   3. Taylor expanded in b around inf

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

   5. Simplified 94.9%

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

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

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

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

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. /-lowering-/.f64 87.9

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

   8. Simplified 87.9%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot \left(b \cdot b\right)}, \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.5% 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 57.8%

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

3. Taylor expanded in b around inf

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

5. Simplified 89.5%

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

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

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

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

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

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

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

   14. unpow2 N/A

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

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

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

   16. /-lowering-/.f64 79.3

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

8. Simplified 79.3%

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

Alternative 5: 81.5% 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 57.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 79.3

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

5. Simplified 79.3%

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

6. Final simplification 79.3%

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

Alternative 6: 64.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 62.1

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

5. Simplified 62.1%

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

   1. neg-sub0 N/A

      \[\leadsto \frac{c}{\color{blue}{0 - b}} \]

   2. flip-- N/A

      \[\leadsto \frac{c}{\color{blue}{\frac{0 \cdot 0 - b \cdot b}{0 + b}}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{c}{\frac{\color{blue}{0} - b \cdot b}{0 + b}} \]

   4. neg-sub0 N/A

      \[\leadsto \frac{c}{\frac{\color{blue}{\mathsf{neg}\left(b \cdot b\right)}}{0 + b}} \]

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

      \[\leadsto \frac{c}{\color{blue}{\frac{\mathsf{neg}\left(b \cdot b\right)}{0 + b}}} \]

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

      \[\leadsto \frac{c}{\frac{\color{blue}{b \cdot \left(\mathsf{neg}\left(b\right)\right)}}{0 + b}} \]

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

      \[\leadsto \frac{c}{\frac{\color{blue}{b \cdot \left(\mathsf{neg}\left(b\right)\right)}}{0 + b}} \]

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

      \[\leadsto \frac{c}{\frac{b \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{0 + b}} \]

   9. +-lowering-+.f64 62.1

      \[\leadsto \frac{c}{\frac{b \cdot \left(-b\right)}{\color{blue}{0 + b}}} \]

7. Applied egg-rr 62.1%

   \[\leadsto \frac{c}{\color{blue}{\frac{b \cdot \left(-b\right)}{0 + b}}} \]

8. Final simplification 62.1%

   \[\leadsto \frac{c}{\frac{b \cdot b}{-b}} \]
9. Add Preprocessing

Alternative 7: 64.6% 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 57.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 62.1

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

5. Simplified 62.1%

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

6. Final simplification 62.1%

   \[\leadsto \frac{-c}{b} \]
7. Add Preprocessing

Alternative 8: 3.2% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 57.8%

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

   1. +-commutative N/A

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

   2. unsub-neg N/A

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

   3. div-sub N/A

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

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

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

4. Applied egg-rr 57.0%

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

   1. sub-neg N/A

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

   2. associate-/r* N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

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

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

   6. /-lowering-/.f64 N/A

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

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

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

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

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

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

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

   14. distribute-neg-frac2 N/A

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

   15. /-lowering-/.f64 N/A

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

   16. *-commutative N/A

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

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

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

   18. metadata-eval N/A

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

6. Applied egg-rr 57.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}{a}, 0.5, \frac{b}{-2 \cdot a}\right)} \]

7. Taylor expanded in c around 0

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.2

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

9. Simplified 3.2%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))