Quadratic roots, narrow range

Percentage Accurate: 55.4% → 99.4%

Time: 15.2s

Alternatives: 5

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: 55.4% 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: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

   1. *-commutative 59.9%

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

3. Simplified 59.9%

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

   1. add-sqr-sqrt 59.9%

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

   2. sqrt-unprod 59.9%

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

   3. pow2 59.9%

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

   4. pow2 59.9%

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

   5. pow-prod-up 59.9%

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

   6. metadata-eval 59.9%

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

6. Applied egg-rr 59.9%

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

   1. flip-+ 59.5%

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

   2. pow2 59.5%

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

   3. sqrt-pow1 59.8%

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

   4. metadata-eval 59.8%

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

   5. sqrt-pow1 59.7%

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

   6. metadata-eval 59.7%

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

   7. add-sqr-sqrt 61.2%

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

   8. associate-*l* 61.2%

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

8. Applied egg-rr 61.2%

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

9. Taylor expanded in b around 0 99.3%

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

    1. associate-*r* 99.3%

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

    2. *-commutative 99.3%

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

    3. *-commutative 99.3%

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

11. Simplified 99.3%

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

    1. div-inv 99.1%

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

    2. associate-/l* 99.2%

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

    3. cancel-sign-sub-inv 99.2%

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

    4. unpow2 99.2%

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

    5. metadata-eval 99.2%

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

    6. fma-define 99.2%

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

    7. *-commutative 99.2%

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

13. Applied egg-rr 99.2%

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

    1. associate-*r/ 99.2%

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

    2. fma-undefine 99.1%

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

    3. unpow2 99.1%

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

    4. +-commutative 99.1%

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

    5. fma-define 99.1%

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

    6. times-frac 99.3%

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

    7. *-rgt-identity 99.3%

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

    8. *-commutative 99.3%

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

    9. associate-*r/ 99.3%

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

    10. associate-/r* 99.4%

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

15. Simplified 99.4%

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

16. Final simplification 99.4%

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

Alternative 2: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.00066:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot \frac{c}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))))
   (if (<= t_0 -0.00066) t_0 (/ 1.0 (/ (- (* a (/ c b)) b) c)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -0.00066) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((a * (c / b)) - b) / 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    if (t_0 <= (-0.00066d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((a * (c / b)) - b) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	double tmp;
	if (t_0 <= -0.00066) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((a * (c / b)) - b) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	tmp = 0
	if t_0 <= -0.00066:
		tmp = t_0
	else:
		tmp = 1.0 / (((a * (c / b)) - b) / c)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a))
	tmp = 0.0
	if (t_0 <= -0.00066)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(a * Float64(c / b)) - b) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.00066)
		tmp = t_0;
	else
		tmp = 1.0 / (((a * (c / b)) - b) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.00066], t$95$0, N[(1.0 / N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -6.6e-4

   1. Initial program 80.2%

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

   if -6.6e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in b around inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

      3. mul-1-neg 88.8%

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

      4. associate-/l* 88.8%

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

   7. Simplified 88.8%

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

      1. clear-num 88.6%

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

      2. inv-pow 88.6%

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

      3. div-inv 88.6%

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

      4. pow-flip 88.6%

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

      5. metadata-eval 88.6%

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

   9. Applied egg-rr 88.6%

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

       1. unpow-1 88.6%

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

       2. cancel-sign-sub-inv 88.6%

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

       3. *-commutative 88.6%

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

       4. +-commutative 88.6%

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

       5. unsub-neg 88.6%

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

       6. distribute-rgt-neg-out 88.6%

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

       7. *-commutative 88.6%

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

       8. distribute-rgt-neg-in 88.6%

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

       9. *-commutative 88.6%

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

       10. distribute-rgt-neg-in 88.6%

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

   11. Simplified 88.6%

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

   12. Taylor expanded in c around 0 89.2%

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

       1. neg-mul-1 89.2%

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

       2. +-commutative 89.2%

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

       3. sub-neg 89.2%

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

       4. associate-/l* 89.2%

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

   14. Simplified 89.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.00066:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot \frac{c}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * (a * 4.0)) / (-b - sqrt(((b * b) - ((c * a) * 4.0))))) / (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 = ((c * (a * 4.0d0)) / (-b - sqrt(((b * b) - ((c * a) * 4.0d0))))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

   1. *-commutative 59.9%

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

3. Simplified 59.9%

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

   1. add-sqr-sqrt 59.9%

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

   2. sqrt-unprod 59.9%

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

   3. pow2 59.9%

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

   4. pow2 59.9%

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

   5. pow-prod-up 59.9%

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

   6. metadata-eval 59.9%

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

6. Applied egg-rr 59.9%

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

   1. flip-+ 59.5%

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

   2. pow2 59.5%

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

   3. sqrt-pow1 59.8%

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

   4. metadata-eval 59.8%

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

   5. sqrt-pow1 59.7%

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

   6. metadata-eval 59.7%

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

   7. add-sqr-sqrt 61.2%

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

   8. associate-*l* 61.2%

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

8. Applied egg-rr 61.2%

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

9. Taylor expanded in b around 0 99.3%

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

    1. associate-*r* 99.3%

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

    2. *-commutative 99.3%

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

    3. *-commutative 99.3%

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

11. Simplified 99.3%

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

    1. unpow2 99.3%

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

13. Applied egg-rr 99.3%

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

14. Final simplification 99.3%

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

Alternative 4: 82.2% accurate, 12.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (b / 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 / ((a / b) - (b / c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.9%

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

   1. *-commutative 59.9%

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

3. Simplified 59.9%

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

5. Taylor expanded in b around inf 77.7%

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

   1. mul-1-neg 77.7%

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

   2. unsub-neg 77.7%

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

   3. mul-1-neg 77.7%

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

   4. associate-/l* 77.7%

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

7. Simplified 77.7%

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

   1. clear-num 77.7%

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

   2. inv-pow 77.7%

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

   3. div-inv 77.7%

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

   4. pow-flip 77.7%

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

   5. metadata-eval 77.7%

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

9. Applied egg-rr 77.7%

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

    1. unpow-1 77.7%

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

    2. cancel-sign-sub-inv 77.7%

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

    3. *-commutative 77.7%

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

    4. +-commutative 77.7%

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

    5. unsub-neg 77.7%

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

    6. distribute-rgt-neg-out 77.7%

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

    7. *-commutative 77.7%

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

    8. distribute-rgt-neg-in 77.7%

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

    9. *-commutative 77.7%

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

    10. distribute-rgt-neg-in 77.7%

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

11. Simplified 77.7%

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

12. Taylor expanded in a around 0 78.5%

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

    1. neg-mul-1 78.5%

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

    2. distribute-frac-neg2 78.5%

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

    3. +-commutative 78.5%

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

    4. distribute-frac-neg2 78.5%

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

    5. unsub-neg 78.5%

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

14. Simplified 78.5%

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

Alternative 5: 64.4% 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 59.9%

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

   1. *-commutative 59.9%

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

3. Simplified 59.9%

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

5. Taylor expanded in b around inf 60.8%

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

   1. associate-*r/ 60.8%

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

   2. mul-1-neg 60.8%

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

7. Simplified 60.8%

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

8. Final simplification 60.8%

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

Reproduce

herbie shell --seed 2024117 
(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)))