Quadratic roots, narrow range

Percentage Accurate: 55.4% → 99.5%

Time: 11.1s

Alternatives: 8

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: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. flip-+ 56.8%

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

   2. pow2 56.8%

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

   3. add-sqr-sqrt 58.3%

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

   4. *-commutative 58.3%

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

   5. *-commutative 58.3%

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

   6. *-commutative 58.3%

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

   7. *-commutative 58.3%

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

3. Applied egg-rr 58.3%

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

4. Taylor expanded in b around 0 99.2%

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

   1. expm1-log1p-u 82.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)\right)} \]

   2. expm1-udef 61.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)} - 1} \]

6. Applied egg-rr 61.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 82.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)\right)} \]

   2. expm1-log1p 99.2%

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

   3. associate-/r* 99.4%

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

   4. *-commutative 99.4%

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

   5. times-frac 99.4%

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

   6. metadata-eval 99.4%

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

   7. sub-neg 99.4%

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

   8. +-commutative 99.4%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-neg-in 99.4%

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

   12. fma-def 99.4%

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

   13. *-commutative 99.4%

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

   14. distribute-rgt-neg-in 99.4%

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

   15. metadata-eval 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in c around 0 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.025:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{a}}{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.025)
   (/ (- (sqrt (+ (* b b) (* -4.0 (* c a)))) b) (* 2.0 a))
   (/ (* 2.0 (/ (* c a) a)) (+ (* 2.0 (/ (* c a) b)) (* b -2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.025) {
		tmp = (sqrt(((b * b) + (-4.0 * (c * a)))) - b) / (2.0 * a);
	} else {
		tmp = (2.0 * ((c * a) / a)) / ((2.0 * ((c * a) / b)) + (b * -2.0));
	}
	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 (((sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)) <= (-0.025d0)) then
        tmp = (sqrt(((b * b) + ((-4.0d0) * (c * a)))) - b) / (2.0d0 * a)
    else
        tmp = (2.0d0 * ((c * a) / a)) / ((2.0d0 * ((c * a) / b)) + (b * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.025) {
		tmp = (Math.sqrt(((b * b) + (-4.0 * (c * a)))) - b) / (2.0 * a);
	} else {
		tmp = (2.0 * ((c * a) / a)) / ((2.0 * ((c * a) / b)) + (b * -2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.025:
		tmp = (math.sqrt(((b * b) + (-4.0 * (c * a)))) - b) / (2.0 * a)
	else:
		tmp = (2.0 * ((c * a) / a)) / ((2.0 * ((c * a) / b)) + (b * -2.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.025)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(c * a) / a)) / Float64(Float64(2.0 * Float64(Float64(c * a) / b)) + Float64(b * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.025)
		tmp = (sqrt(((b * b) + (-4.0 * (c * a)))) - b) / (2.0 * a);
	else
		tmp = (2.0 * ((c * a) / a)) / ((2.0 * ((c * a) / b)) + (b * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -0.025], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -0.025000000000000001

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. +-commutative 81.9%

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

      3. unsub-neg 81.9%

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

      4. fma-neg 82.0%

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

      5. associate-*l* 82.0%

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

      6. *-commutative 82.0%

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

      7. distribute-rgt-neg-in 82.0%

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

      8. metadata-eval 82.0%

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

   3. Simplified 82.0%

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

      1. fma-udef 81.9%

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

      2. *-commutative 81.9%

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

   5. Applied egg-rr 81.9%

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

   if -0.025000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 47.0%

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

      1. flip-+ 47.2%

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

      2. pow2 47.1%

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

      3. add-sqr-sqrt 48.5%

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

      4. *-commutative 48.5%

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

      5. *-commutative 48.5%

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

      6. *-commutative 48.5%

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

      7. *-commutative 48.5%

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

   3. Applied egg-rr 48.5%

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

   4. Taylor expanded in b around 0 99.2%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)\right)} \]

      2. expm1-udef 70.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)} - 1} \]

   6. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 99.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)\right)} \]

      2. expm1-log1p 99.2%

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

      3. associate-/r* 99.4%

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

      4. *-commutative 99.4%

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

      5. times-frac 99.4%

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

      6. metadata-eval 99.4%

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

      7. sub-neg 99.4%

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

      8. +-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-neg-in 99.4%

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

      12. fma-def 99.4%

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

      13. *-commutative 99.4%

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

      14. distribute-rgt-neg-in 99.4%

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

      15. metadata-eval 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in b around inf 88.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.025:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{a}}{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. flip-+ 56.8%

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

   2. pow2 56.8%

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

   3. add-sqr-sqrt 58.3%

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

   4. *-commutative 58.3%

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

   5. *-commutative 58.3%

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

   6. *-commutative 58.3%

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

   7. *-commutative 58.3%

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

3. Applied egg-rr 58.3%

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

4. Taylor expanded in b around 0 99.2%

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

   1. expm1-log1p-u 82.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)\right)} \]

   2. expm1-udef 61.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)} - 1} \]

6. Applied egg-rr 61.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 82.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)\right)} \]

   2. expm1-log1p 99.2%

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

   3. associate-/r* 99.4%

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

   4. *-commutative 99.4%

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

   5. times-frac 99.4%

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

   6. metadata-eval 99.4%

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

   7. sub-neg 99.4%

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

   8. +-commutative 99.4%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-neg-in 99.4%

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

   12. fma-def 99.4%

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

   13. *-commutative 99.4%

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

   14. distribute-rgt-neg-in 99.4%

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

   15. metadata-eval 99.4%

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

8. Simplified 99.4%

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

   1. fma-udef 99.4%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6:\\ \;\;\;\;\left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{a}}{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.6)
   (* (- (sqrt (+ (* b b) (* -4.0 (* c a)))) b) (/ 0.5 a))
   (/ (* 2.0 (/ (* c a) a)) (+ (* 2.0 (/ (* c a) b)) (* b -2.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.6) {
		tmp = (sqrt(((b * b) + (-4.0 * (c * a)))) - b) * (0.5 / a);
	} else {
		tmp = (2.0 * ((c * a) / a)) / ((2.0 * ((c * a) / b)) + (b * -2.0));
	}
	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 <= 7.6d0) then
        tmp = (sqrt(((b * b) + ((-4.0d0) * (c * a)))) - b) * (0.5d0 / a)
    else
        tmp = (2.0d0 * ((c * a) / a)) / ((2.0d0 * ((c * a) / b)) + (b * (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.6) {
		tmp = (Math.sqrt(((b * b) + (-4.0 * (c * a)))) - b) * (0.5 / a);
	} else {
		tmp = (2.0 * ((c * a) / a)) / ((2.0 * ((c * a) / b)) + (b * -2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.6:
		tmp = (math.sqrt(((b * b) + (-4.0 * (c * a)))) - b) * (0.5 / a)
	else:
		tmp = (2.0 * ((c * a) / a)) / ((2.0 * ((c * a) / b)) + (b * -2.0))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(-4.0 * Float64(c * a)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(c * a) / a)) / Float64(Float64(2.0 * Float64(Float64(c * a) / b)) + Float64(b * -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.6)
		tmp = (sqrt(((b * b) + (-4.0 * (c * a)))) - b) * (0.5 / a);
	else
		tmp = (2.0 * ((c * a) / a)) / ((2.0 * ((c * a) / b)) + (b * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5999999999999996

   1. Initial program 84.1%

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

      1. /-rgt-identity 84.1%

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

      2. metadata-eval 84.1%

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

      3. associate-/l* 84.1%

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

      4. associate-*r/ 84.0%

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

      5. +-commutative 84.0%

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

      6. unsub-neg 84.0%

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

      7. fma-neg 84.3%

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

      8. associate-*l* 84.3%

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

      9. *-commutative 84.3%

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

      10. distribute-rgt-neg-in 84.3%

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

      11. metadata-eval 84.3%

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

      12. associate-/r* 84.3%

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

      13. metadata-eval 84.3%

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

      14. metadata-eval 84.3%

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

   3. Simplified 84.3%

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

      1. fma-udef 84.0%

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

      2. *-commutative 84.0%

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

   5. Applied egg-rr 84.0%

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

   if 7.5999999999999996 < b

   1. Initial program 49.4%

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

      1. flip-+ 49.4%

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

      2. pow2 49.4%

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

      3. add-sqr-sqrt 50.9%

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

      4. *-commutative 50.9%

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

      5. *-commutative 50.9%

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

      6. *-commutative 50.9%

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

      7. *-commutative 50.9%

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

   3. Applied egg-rr 50.9%

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

   4. Taylor expanded in b around 0 99.2%

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

      1. expm1-log1p-u 85.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)\right)} \]

      2. expm1-udef 62.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)} - 1} \]

   6. Applied egg-rr 62.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 85.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)\right)} \]

      2. expm1-log1p 99.2%

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

      3. associate-/r* 99.3%

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

      4. *-commutative 99.3%

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

      5. times-frac 99.3%

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

      6. metadata-eval 99.3%

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

      7. sub-neg 99.3%

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

      8. +-commutative 99.3%

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

      9. associate-*r* 99.3%

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

      10. *-commutative 99.3%

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

      11. distribute-rgt-neg-in 99.3%

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

      12. fma-def 99.4%

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

      13. *-commutative 99.4%

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

      14. distribute-rgt-neg-in 99.4%

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

      15. metadata-eval 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in b around inf 86.3%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6:\\ \;\;\;\;\left(\sqrt{b \cdot b + -4 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{a}}{2 \cdot \frac{c \cdot a}{b} + b \cdot -2}\\ \end{array} \]

Alternative 5: 82.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b, c):
	return (2.0 * ((c * a) / a)) / ((2.0 * ((c * a) / b)) + (b * -2.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. flip-+ 56.8%

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

   2. pow2 56.8%

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

   3. add-sqr-sqrt 58.3%

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

   4. *-commutative 58.3%

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

   5. *-commutative 58.3%

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

   6. *-commutative 58.3%

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

   7. *-commutative 58.3%

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

3. Applied egg-rr 58.3%

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

4. Taylor expanded in b around 0 99.2%

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

   1. expm1-log1p-u 82.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)\right)} \]

   2. expm1-udef 61.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2 \cdot a}\right)} - 1} \]

6. Applied egg-rr 61.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 82.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\right)\right)} \]

   2. expm1-log1p 99.2%

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

   3. associate-/r* 99.4%

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

   4. *-commutative 99.4%

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

   5. times-frac 99.4%

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

   6. metadata-eval 99.4%

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

   7. sub-neg 99.4%

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

   8. +-commutative 99.4%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-neg-in 99.4%

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

   12. fma-def 99.4%

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

   13. *-commutative 99.4%

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

   14. distribute-rgt-neg-in 99.4%

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

   15. metadata-eval 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in b around inf 79.9%

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

10. Final simplification 79.9%

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

Alternative 6: 81.9% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. neg-sub0 56.8%

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

   2. associate-+l- 56.8%

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

   3. sub0-neg 56.8%

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

   4. neg-mul-1 56.8%

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

   5. associate-*l/ 56.8%

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

   6. *-commutative 56.8%

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

   7. associate-/r* 56.8%

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

   8. /-rgt-identity 56.8%

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

   9. metadata-eval 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in b around inf 79.2%

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

   1. +-commutative 79.2%

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

   2. mul-1-neg 79.2%

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

   3. unsub-neg 79.2%

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

   4. associate-*r/ 79.2%

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

   5. neg-mul-1 79.2%

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

   6. unpow2 79.2%

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

   7. associate-*l* 79.2%

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

6. Simplified 79.2%

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

   1. unpow3 79.2%

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

8. Applied egg-rr 79.2%

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

9. Final simplification 79.2%

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

Alternative 7: 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(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 56.8%

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

   1. neg-sub0 56.8%

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

   2. associate-+l- 56.8%

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

   3. sub0-neg 56.8%

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

   4. neg-mul-1 56.8%

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

   5. associate-*l/ 56.8%

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

   6. *-commutative 56.8%

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

   7. associate-/r* 56.8%

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

   8. /-rgt-identity 56.8%

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

   9. metadata-eval 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in b around inf 63.3%

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

   1. associate-*r/ 63.3%

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

   2. neg-mul-1 63.3%

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

6. Simplified 63.3%

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

7. Final simplification 63.3%

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

Alternative 8: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

   1. add-cube-cbrt 56.8%

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

   2. pow3 56.8%

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

   3. neg-mul-1 56.8%

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

   4. fma-def 56.8%

      \[\leadsto {\left(\sqrt[3]{\frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}}\right)}^{3} \]

   5. *-commutative 56.8%

      \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}\right)}{2 \cdot a}}\right)}^{3} \]

   6. *-commutative 56.8%

      \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}\right)}{2 \cdot a}}\right)}^{3} \]

   7. *-commutative 56.8%

      \[\leadsto {\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{\color{blue}{a \cdot 2}}}\right)}^{3} \]

3. Applied egg-rr 56.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right)}{a \cdot 2}}\right)}^{3}} \]

4. Taylor expanded in c around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{b + -1 \cdot b}{a} \cdot {1}^{0.3333333333333333}\right)} \]
5. Step-by-step derivation

   1. distribute-rgt1-in 3.2%

      \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\left(-1 + 1\right) \cdot b}}{a} \cdot {1}^{0.3333333333333333}\right) \]

   2. metadata-eval 3.2%

      \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{0} \cdot b}{a} \cdot {1}^{0.3333333333333333}\right) \]

   3. mul0-lft 3.2%

      \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{0}}{a} \cdot {1}^{0.3333333333333333}\right) \]

   4. pow-base-1 3.2%

      \[\leadsto 0.5 \cdot \left(\frac{0}{a} \cdot \color{blue}{1}\right) \]

   5. associate-*l/ 3.2%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{0 \cdot 1}{a}} \]

   6. metadata-eval 3.2%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{0}}{a} \]

   7. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.5 \cdot 0}{a}} \]

   8. metadata-eval 3.2%

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

6. Simplified 3.2%

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

7. Final simplification 3.2%

   \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023175 
(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)))