Quadratic roots, wide range

Percentage Accurate: 17.8% → 99.7%

Time: 10.3s

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: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 17.3

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   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 17.3

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

4. Applied rewrites 17.3%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]
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. lift-+.f64 N/A

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

   7. frac-times N/A

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

   8. 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 99.4%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, 0\right)}{a \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, 0\right)}{\color{blue}{a \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, 0\right)}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, 0\right)}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}} \]

   5. lower-/.f64 99.7

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

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, 0\right)}}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b} \]

   7. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{1}{2} \cdot \color{blue}{\left(\left(-4 \cdot c\right) \cdot a + 0\right)}}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b} \]

   8. +-rgt-identity N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 17.3

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   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 17.3

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

4. Applied rewrites 17.3%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]
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. lift-+.f64 N/A

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

   7. frac-times N/A

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

   8. 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 99.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \mathsf{fma}\left(-4 \cdot c, a, 0\right)}}{a \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left(\left(-4 \cdot c\right) \cdot a + 0\right)}}{a \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)} \]

   3. +-rgt-identity N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.4

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.4

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

   16. lift-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 99.4

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 3: 95.2% accurate, 1.2× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

3. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. associate-*r/ N/A

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

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. div-sub N/A

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

   8. unsub-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-out N/A

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

   11. associate-/l* N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-/.f64 N/A

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

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

8. Final simplification 95.5%

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

Alternative 4: 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(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 17.3%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 90.7

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

5. Applied rewrites 90.7%

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

Alternative 5: 3.3% 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 17.3%

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

4. Applied rewrites 14.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

6. Applied rewrites 19.1%

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

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 3.3

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

9. Applied rewrites 3.3%

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

Reproduce

herbie shell --seed 2024282 
(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)))