quadm (p42, negative)

Percentage Accurate: 52.4% → 85.4%

Time: 8.8s

Alternatives: 11

Speedup: 2.5×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{\left(-2\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-82)
   (pow (- (/ a b) (/ b c)) -1.0)
   (if (<= b 5.2e+82)
     (/ (+ (sqrt (fma -4.0 (* c a) (* b b))) b) (* (- 2.0) a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-82) {
		tmp = pow(((a / b) - (b / c)), -1.0);
	} else if (b <= 5.2e+82) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.3e-82)
		tmp = Float64(Float64(a / b) - Float64(b / c)) ^ -1.0;
	elseif (b <= 5.2e+82)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b) / Float64(Float64(-2.0) * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.3e-82], N[Power[N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[b, 5.2e+82], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[((-2.0) * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.30000000000000019e-82

   1. Initial program 19.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 19.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 19.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 19.7

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

   4. Applied rewrites 19.7%

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

   6. Applied rewrites 19.7%

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

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{a}{{b}^{2}} + \frac{1}{c}\right)\right)}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.4

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

   9. Applied rewrites 88.4%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 88.5%

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

   if -4.30000000000000019e-82 < b < 5.1999999999999997e82

   1. Initial program 81.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 81.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 81.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 81.7

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

   4. Applied rewrites 81.7%

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

   if 5.1999999999999997e82 < b

   1. Initial program 55.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.5

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

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{\left(-2\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-82)
   (pow (- (/ a b) (/ b c)) -1.0)
   (if (<= b 5.2e+82)
     (/ (+ (sqrt (fma (* -4.0 a) c (* b b))) b) (* -2.0 a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-82) {
		tmp = pow(((a / b) - (b / c)), -1.0);
	} else if (b <= 5.2e+82) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + b) / (-2.0 * a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.3e-82)
		tmp = Float64(Float64(a / b) - Float64(b / c)) ^ -1.0;
	elseif (b <= 5.2e+82)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.3e-82], N[Power[N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[b, 5.2e+82], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.30000000000000019e-82

   1. Initial program 19.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 19.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 19.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 19.7

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

   4. Applied rewrites 19.7%

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

   6. Applied rewrites 19.7%

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

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{a}{{b}^{2}} + \frac{1}{c}\right)\right)}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.4

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

   9. Applied rewrites 88.4%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 88.5%

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

   if -4.30000000000000019e-82 < b < 5.1999999999999997e82

   1. Initial program 81.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 81.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 81.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 81.7

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

   4. Applied rewrites 81.7%

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

   6. Applied rewrites 81.7%

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

   if 5.1999999999999997e82 < b

   1. Initial program 55.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.5

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

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-82)
   (pow (- (/ a b) (/ b c)) -1.0)
   (if (<= b 5.2e+82)
     (* (/ -0.5 a) (+ (sqrt (fma (* c -4.0) a (* b b))) b))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-82) {
		tmp = pow(((a / b) - (b / c)), -1.0);
	} else if (b <= 5.2e+82) {
		tmp = (-0.5 / a) * (sqrt(fma((c * -4.0), a, (b * b))) + b);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.3e-82)
		tmp = Float64(Float64(a / b) - Float64(b / c)) ^ -1.0;
	elseif (b <= 5.2e+82)
		tmp = Float64(Float64(-0.5 / a) * Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) + b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.3e-82], N[Power[N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[b, 5.2e+82], N[(N[(-0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.30000000000000019e-82

   1. Initial program 19.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 19.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 19.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 19.7

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

   4. Applied rewrites 19.7%

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

   6. Applied rewrites 19.7%

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

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{a}{{b}^{2}} + \frac{1}{c}\right)\right)}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.4

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

   9. Applied rewrites 88.4%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 88.5%

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

   if -4.30000000000000019e-82 < b < 5.1999999999999997e82

   1. Initial program 81.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 81.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 81.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 81.7

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

   4. Applied rewrites 81.7%

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

   6. Applied rewrites 81.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 81.4

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lift-*.f64 81.4

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

   8. Applied rewrites 81.4%

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

   if 5.1999999999999997e82 < b

   1. Initial program 55.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.5

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

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-82)
   (pow (- (/ a b) (/ b c)) -1.0)
   (if (<= b 3.4e-92)
     (/ (+ (sqrt (* (* c a) -4.0)) b) (* -2.0 a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-82) {
		tmp = pow(((a / b) - (b / c)), -1.0);
	} else if (b <= 3.4e-92) {
		tmp = (sqrt(((c * a) * -4.0)) + b) / (-2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.3d-82)) then
        tmp = ((a / b) - (b / c)) ** (-1.0d0)
    else if (b <= 3.4d-92) then
        tmp = (sqrt(((c * a) * (-4.0d0))) + b) / ((-2.0d0) * a)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-82) {
		tmp = Math.pow(((a / b) - (b / c)), -1.0);
	} else if (b <= 3.4e-92) {
		tmp = (Math.sqrt(((c * a) * -4.0)) + b) / (-2.0 * a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.3e-82:
		tmp = math.pow(((a / b) - (b / c)), -1.0)
	elif b <= 3.4e-92:
		tmp = (math.sqrt(((c * a) * -4.0)) + b) / (-2.0 * a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.3e-82)
		tmp = Float64(Float64(a / b) - Float64(b / c)) ^ -1.0;
	elseif (b <= 3.4e-92)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) + b) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.3e-82)
		tmp = ((a / b) - (b / c)) ^ -1.0;
	elseif (b <= 3.4e-92)
		tmp = (sqrt(((c * a) * -4.0)) + b) / (-2.0 * a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.3e-82], N[Power[N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[b, 3.4e-92], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.30000000000000019e-82

   1. Initial program 19.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 19.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 19.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 19.7

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

   4. Applied rewrites 19.7%

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

   6. Applied rewrites 19.7%

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

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{a}{{b}^{2}} + \frac{1}{c}\right)\right)}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.4

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

   9. Applied rewrites 88.4%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 88.5%

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

   if -4.30000000000000019e-82 < b < 3.4000000000000003e-92

   1. Initial program 72.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 72.9

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 72.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 72.9

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

   4. Applied rewrites 72.9%

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

   6. Applied rewrites 72.9%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

   if 3.4000000000000003e-92 < b

   1. Initial program 71.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.3

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

   5. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} + b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.3e-82)
   (pow (- (/ a b) (/ b c)) -1.0)
   (if (<= b 3.4e-92)
     (* (/ 0.5 a) (- b (sqrt (* -4.0 (* a c)))))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-82) {
		tmp = pow(((a / b) - (b / c)), -1.0);
	} else if (b <= 3.4e-92) {
		tmp = (0.5 / a) * (b - sqrt((-4.0 * (a * c))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.3d-82)) then
        tmp = ((a / b) - (b / c)) ** (-1.0d0)
    else if (b <= 3.4d-92) then
        tmp = (0.5d0 / a) * (b - sqrt(((-4.0d0) * (a * c))))
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.3e-82) {
		tmp = Math.pow(((a / b) - (b / c)), -1.0);
	} else if (b <= 3.4e-92) {
		tmp = (0.5 / a) * (b - Math.sqrt((-4.0 * (a * c))));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.3e-82:
		tmp = math.pow(((a / b) - (b / c)), -1.0)
	elif b <= 3.4e-92:
		tmp = (0.5 / a) * (b - math.sqrt((-4.0 * (a * c))))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.3e-82)
		tmp = Float64(Float64(a / b) - Float64(b / c)) ^ -1.0;
	elseif (b <= 3.4e-92)
		tmp = Float64(Float64(0.5 / a) * Float64(b - sqrt(Float64(-4.0 * Float64(a * c)))));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.3e-82)
		tmp = ((a / b) - (b / c)) ^ -1.0;
	elseif (b <= 3.4e-92)
		tmp = (0.5 / a) * (b - sqrt((-4.0 * (a * c))));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.3e-82], N[Power[N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[b, 3.4e-92], N[(N[(0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.30000000000000019e-82

   1. Initial program 19.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 19.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 19.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 19.7

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

   4. Applied rewrites 19.7%

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

   6. Applied rewrites 19.7%

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

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{a}{{b}^{2}} + \frac{1}{c}\right)\right)}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.4

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

   9. Applied rewrites 88.4%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 88.5%

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

   if -4.30000000000000019e-82 < b < 3.4000000000000003e-92

   1. Initial program 72.9%

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in a around inf

      \[\leadsto \frac{\frac{1}{2}}{a} \cdot \left(b - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 69.7

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

   7. Applied rewrites 69.7%

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

   if 3.4000000000000003e-92 < b

   1. Initial program 71.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.3

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

   5. Applied rewrites 87.3%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-92}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b - \sqrt{-4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{-261}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.95e-261) (pow (- (/ a b) (/ b c)) -1.0) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.95e-261) {
		tmp = pow(((a / b) - (b / c)), -1.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.95d-261) then
        tmp = ((a / b) - (b / c)) ** (-1.0d0)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.95e-261) {
		tmp = Math.pow(((a / b) - (b / c)), -1.0);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.95e-261:
		tmp = math.pow(((a / b) - (b / c)), -1.0)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.95e-261)
		tmp = Float64(Float64(a / b) - Float64(b / c)) ^ -1.0;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.95e-261)
		tmp = ((a / b) - (b / c)) ^ -1.0;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.95e-261], N[Power[N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.95000000000000009e-261

   1. Initial program 36.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-neg.f64 36.2

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 36.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 36.2

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

   4. Applied rewrites 36.2%

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

   6. Applied rewrites 36.1%

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

   7. Taylor expanded in b around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{a}{{b}^{2}} + \frac{1}{c}\right)\right)}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 65.7%

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

   if 1.95000000000000009e-261 < b

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{-261}:\\ \;\;\;\;{\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ c (- b)) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = c / -b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = c / -b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = c / -b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = c / -b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(c / (-b)), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 35.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 71.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 73.2

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

   5. Applied rewrites 73.2%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.55 \cdot 10^{-301}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.55e-301) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.55e-301) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.55d-301)) then
        tmp = c / -b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.55e-301) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.55e-301:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.55e-301)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.55e-301)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.55e-301], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.55e-301

   1. Initial program 33.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   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. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if -3.55e-301 < b

   1. Initial program 72.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.0

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

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 35.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
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. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 32.5

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

5. Applied rewrites 32.5%

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

Alternative 10: 11.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(c / 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 54.4%

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

4. Applied rewrites 32.7%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 12.4

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

7. Applied rewrites 12.4%

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

Alternative 11: 2.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

4. Applied rewrites 32.7%

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

5. Taylor expanded in b around -inf

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

   1. lower-/.f64 2.3

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

7. Applied rewrites 2.3%

   \[\leadsto \color{blue}{\frac{b}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024306 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))