Quadratic roots, narrow range

Percentage Accurate: 55.3% → 99.3%

Time: 13.6s

Alternatives: 7

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. *-commutative 54.6%

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

3. Simplified 54.6%

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

   1. add-cube-cbrt 52.5%

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

   2. pow3 52.5%

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

   3. cbrt-prod 51.6%

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

   4. pow2 51.6%

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

6. Applied egg-rr 51.6%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{3}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. flip-+ 51.6%

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

   2. pow2 51.6%

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

   3. add-sqr-sqrt 51.8%

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

   4. pow-pow 51.7%

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

   5. pow1/3 52.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({\color{blue}{\left({b}^{0.3333333333333333}\right)}}^{\left(2 \cdot 3\right)} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{3} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   6. pow-pow 56.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(0.3333333333333333 \cdot \left(2 \cdot 3\right)\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{3} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   7. metadata-eval 56.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\left(0.3333333333333333 \cdot \color{blue}{6}\right)} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{{\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{3} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. metadata-eval 56.3%

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

   9. *-commutative 56.3%

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

   10. *-commutative 56.3%

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

8. Applied egg-rr 56.3%

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

   1. associate--r- 99.3%

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

   2. unpow2 99.3%

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

   3. unpow2 99.3%

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

   4. difference-of-squares 99.3%

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

   5. remove-double-neg 99.3%

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

   6. distribute-neg-in 99.3%

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

   7. *-lft-identity 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-in 99.3%

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

   10. metadata-eval 99.3%

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

   11. distribute-lft-neg-in 99.3%

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

   12. mul0-rgt 99.3%

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

   13. unpow2 99.3%

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

   14. fma-neg 99.3%

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

   15. associate-*r* 99.3%

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

   16. *-commutative 99.3%

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

   17. distribute-rgt-neg-in 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

Alternative 2: 85.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 24:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{\frac{a}{{b}^{2}} + \frac{-1}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 24.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ (/ 1.0 b) (+ (/ a (pow b 2.0)) (/ -1.0 c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 24.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / b) / ((a / pow(b, 2.0)) + (-1.0 / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 24.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / b) / Float64(Float64(a / (b ^ 2.0)) + Float64(-1.0 / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 24.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] / N[(N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 24

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. +-commutative 80.0%

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

      3. sqr-neg 80.0%

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

      4. unsub-neg 80.0%

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

      5. sqr-neg 80.0%

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

      6. fma-neg 80.1%

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

      7. distribute-lft-neg-in 80.1%

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

      8. *-commutative 80.1%

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

      9. *-commutative 80.1%

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

      10. distribute-rgt-neg-in 80.1%

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

      11. metadata-eval 80.1%

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

   3. Simplified 80.1%

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

   if 24 < b

   1. Initial program 44.8%

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

      1. *-commutative 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in b around inf 88.4%

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

      1. mul-1-neg 88.4%

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

      2. unsub-neg 88.4%

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

      3. mul-1-neg 88.4%

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

      4. associate-/l* 88.4%

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

   7. Simplified 88.4%

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

      1. clear-num 88.2%

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

      2. inv-pow 88.2%

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

      3. div-inv 88.2%

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

      4. pow-flip 88.2%

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

      5. metadata-eval 88.2%

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

   9. Applied egg-rr 88.2%

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

   10. Taylor expanded in b around inf 88.7%

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

       1. *-un-lft-identity 88.7%

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

       2. unpow-1 88.7%

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

   12. Applied egg-rr 88.7%

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

       1. *-lft-identity 88.7%

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

       2. associate-/r* 88.8%

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

       3. sub-neg 88.8%

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

       4. distribute-neg-frac 88.8%

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

       5. metadata-eval 88.8%

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

   14. Simplified 88.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{b}}{\frac{a}{{b}^{2}} + \frac{-1}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{\frac{a}{{b}^{2}} + \frac{-1}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.8)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (/ 1.0 b) (+ (/ a (pow b 2.0)) (/ -1.0 c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / b) / ((a / pow(b, 2.0)) + (-1.0 / c));
	}
	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 <= 4.8d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (1.0d0 / b) / ((a / (b ** 2.0d0)) + ((-1.0d0) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / b) / ((a / Math.pow(b, 2.0)) + (-1.0 / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.8:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = (1.0 / b) / ((a / math.pow(b, 2.0)) + (-1.0 / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / b) / Float64(Float64(a / (b ^ 2.0)) + Float64(-1.0 / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.8)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = (1.0 / b) / ((a / (b ^ 2.0)) + (-1.0 / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] / N[(N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 81.4%

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

   if 4.79999999999999982 < b

   1. Initial program 46.5%

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

      1. *-commutative 46.5%

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

   3. Simplified 46.5%

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

   5. Taylor expanded in b around inf 87.5%

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

      1. mul-1-neg 87.5%

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

      2. unsub-neg 87.5%

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

      3. mul-1-neg 87.5%

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

      4. associate-/l* 87.5%

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

   7. Simplified 87.5%

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

      1. clear-num 87.3%

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

      2. inv-pow 87.3%

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

      3. div-inv 87.3%

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

      4. pow-flip 87.3%

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

      5. metadata-eval 87.3%

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

   9. Applied egg-rr 87.3%

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

   10. Taylor expanded in b around inf 87.8%

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

       1. *-un-lft-identity 87.8%

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

       2. unpow-1 87.8%

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

   12. Applied egg-rr 87.8%

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

       1. *-lft-identity 87.8%

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

       2. associate-/r* 87.9%

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

       3. sub-neg 87.9%

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

       4. distribute-neg-frac 87.9%

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

       5. metadata-eval 87.9%

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

   14. Simplified 87.9%

       \[\leadsto \color{blue}{\frac{\frac{1}{b}}{\frac{a}{{b}^{2}} + \frac{-1}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{\frac{a}{{b}^{2}} + \frac{-1}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{b}}{\frac{a}{{b}^{2}} + \frac{-1}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ 1.0 b) (+ (/ a (pow b 2.0)) (/ -1.0 c))))

C

double code(double a, double b, double c) {
	return (1.0 / b) / ((a / pow(b, 2.0)) + (-1.0 / c));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (1.0d0 / b) / ((a / (b ** 2.0d0)) + ((-1.0d0) / c))
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / b) / ((a / Math.pow(b, 2.0)) + (-1.0 / c));
}

Python

def code(a, b, c):
	return (1.0 / b) / ((a / math.pow(b, 2.0)) + (-1.0 / c))

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / b) / Float64(Float64(a / (b ^ 2.0)) + Float64(-1.0 / c)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / b) / ((a / (b ^ 2.0)) + (-1.0 / c));
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / b), $MachinePrecision] / N[(N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. *-commutative 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in b around inf 81.3%

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

   1. mul-1-neg 81.3%

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

   2. unsub-neg 81.3%

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

   3. mul-1-neg 81.3%

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

   4. associate-/l* 81.3%

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

7. Simplified 81.3%

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

   1. clear-num 81.1%

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

   2. inv-pow 81.1%

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

   3. div-inv 81.1%

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

   4. pow-flip 81.1%

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

   5. metadata-eval 81.1%

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

9. Applied egg-rr 81.1%

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

10. Taylor expanded in b around inf 81.8%

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

    1. *-un-lft-identity 81.8%

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

    2. unpow-1 81.8%

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

12. Applied egg-rr 81.8%

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

    1. *-lft-identity 81.8%

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

    2. associate-/r* 81.9%

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

    3. sub-neg 81.9%

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

    4. distribute-neg-frac 81.9%

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

    5. metadata-eval 81.9%

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

14. Simplified 81.9%

    \[\leadsto \color{blue}{\frac{\frac{1}{b}}{\frac{a}{{b}^{2}} + \frac{-1}{c}}} \]
15. Add Preprocessing

Alternative 5: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{a}{b} - \frac{b}{c}\right)}^{-1} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((a / b) - (b / c)) ** (-1.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return Math.pow(((a / b) - (b / c)), -1.0);
}

Python

def code(a, b, c):
	return math.pow(((a / b) - (b / c)), -1.0)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((a / b) - (b / c)) ^ -1.0;
end

Wolfram

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

Derivation

1. Initial program 54.6%

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

   1. *-commutative 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in b around inf 81.3%

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

   1. mul-1-neg 81.3%

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

   2. unsub-neg 81.3%

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

   3. mul-1-neg 81.3%

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

   4. associate-/l* 81.3%

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

7. Simplified 81.3%

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

   1. clear-num 81.1%

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

   2. inv-pow 81.1%

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

   3. div-inv 81.1%

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

   4. pow-flip 81.1%

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

   5. metadata-eval 81.1%

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

9. Applied egg-rr 81.1%

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

10. Taylor expanded in a around 0 81.9%

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

    1. +-commutative 81.9%

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

    2. mul-1-neg 81.9%

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

    3. unsub-neg 81.9%

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

12. Simplified 81.9%

    \[\leadsto {\color{blue}{\left(\frac{a}{b} - \frac{b}{c}\right)}}^{-1} \]
13. Add Preprocessing

Alternative 6: 64.5% accurate, 29.0× 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 54.6%

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

   1. *-commutative 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in b around inf 64.5%

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

   1. associate-*r/ 64.5%

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

   2. mul-1-neg 64.5%

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

7. Simplified 64.5%

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

8. Final simplification 64.5%

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

Alternative 7: 3.2% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 54.6%

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

   1. *-commutative 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in b around inf 64.5%

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

   1. associate-*r/ 64.5%

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

   2. mul-1-neg 64.5%

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

7. Simplified 64.5%

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

   1. div-inv 64.4%

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

9. Applied egg-rr 64.4%

   \[\leadsto \color{blue}{\left(-c\right) \cdot \frac{1}{b}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 56.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-c\right) \cdot \frac{1}{b}\right)\right)} \]

    2. expm1-undefine 42.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-c\right) \cdot \frac{1}{b}\right)} - 1} \]

    3. distribute-lft-neg-out 42.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-c \cdot \frac{1}{b}}\right)} - 1 \]

    4. div-inv 42.4%

       \[\leadsto e^{\mathsf{log1p}\left(-\color{blue}{\frac{c}{b}}\right)} - 1 \]

11. Applied egg-rr 42.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{c}{b}\right)} - 1} \]
12. Step-by-step derivation

    1. sub-neg 42.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{c}{b}\right)} + \left(-1\right)} \]

    2. log1p-undefine 42.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + \left(-\frac{c}{b}\right)\right)}} + \left(-1\right) \]

    3. rem-exp-log 50.7%

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

    4. unsub-neg 50.7%

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

    5. metadata-eval 50.7%

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

13. Simplified 50.7%

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

14. Taylor expanded in c around 0 3.2%

    \[\leadsto \color{blue}{1} + -1 \]

15. Final simplification 3.2%

    \[\leadsto 0 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))