Quadratic roots, narrow range

Percentage Accurate: 55.5% → 92.2%

Time: 11.4s

Alternatives: 7

Speedup: 29.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% 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: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -36:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{20}{a}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -36.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (+
    (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (-
     (-
      (* -0.25 (* (/ (pow (* a c) 4.0) (pow b 7.0)) (/ 20.0 a)))
      (/ (* a (pow c 2.0)) (pow b 3.0)))
     (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -36.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * ((pow((a * c), 4.0) / pow(b, 7.0)) * (20.0 / a))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -36.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(20.0 / a))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -36.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(20.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $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)) < -36

   1. Initial program 91.0%

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

   3. Simplified 91.2%

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

   if -36 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 53.8%

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

      1. *-commutative 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in b around inf 93.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -2\right)}}^{2}}{a \cdot {b}^{7}}\right)\right) \]

      2. unpow-prod-down 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left({a}^{2} \cdot {c}^{2}\right)}^{2} \cdot {-2}^{2}}}{a \cdot {b}^{7}}\right)\right) \]

      3. pow-prod-down 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\color{blue}{\left({\left(a \cdot c\right)}^{2}\right)}}^{2} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

      4. pow-pow 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

      5. metadata-eval 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{\color{blue}{4}} \cdot {-2}^{2}}{a \cdot {b}^{7}}\right)\right) \]

      6. metadata-eval 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{4} \cdot \color{blue}{4}}{a \cdot {b}^{7}}\right)\right) \]

   6. Applied egg-rr 93.0%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{4} \cdot 4}}{a \cdot {b}^{7}}\right)\right) \]

   7. Taylor expanded in c around 0 93.0%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
   8. Step-by-step derivation

      1. distribute-rgt-out 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(4 + 16\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

      2. associate-*r* 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(4 + 16\right)}}{a \cdot {b}^{7}}\right)\right) \]

      3. *-commutative 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(4 + 16\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. *-commutative 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot \left(4 + 16\right)}{\color{blue}{{b}^{7} \cdot a}}\right)\right) \]

      5. times-frac 93.0%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{{b}^{7}} \cdot \frac{4 + 16}{a}\right)}\right)\right) \]

   9. Simplified 93.0%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{20}{a}\right)}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -36:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{20}{a}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 2: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.35:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.35)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (- (/ (* -2.0 (* (pow a 2.0) (pow c 3.0))) (pow b 5.0)) (/ c b))
    (/ a (/ (/ (pow b 3.0) c) c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.35) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = (((-2.0 * (pow(a, 2.0) * pow(c, 3.0))) / pow(b, 5.0)) - (c / b)) - (a / ((pow(b, 3.0) / c) / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.35)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(-2.0 * Float64((a ^ 2.0) * (c ^ 3.0))) / (b ^ 5.0)) - Float64(c / b)) - Float64(a / Float64(Float64((b ^ 3.0) / c) / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.35], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision] / c), $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)) < -0.34999999999999998

   1. Initial program 84.9%

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

   3. Simplified 85.2%

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

   if -0.34999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in b around inf 92.1%

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

      1. associate-+r+ 92.1%

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{c}{b}\right) + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      2. mul-1-neg 92.1%

         \[\leadsto \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{c}{b}\right) + \color{blue}{\left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

      3. unsub-neg 92.1%

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{c}{b}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      4. mul-1-neg 92.1%

         \[\leadsto \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \color{blue}{\left(-\frac{c}{b}\right)}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      5. unsub-neg 92.1%

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \frac{c}{b}\right)} - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      6. associate-*r/ 92.1%

         \[\leadsto \left(\color{blue}{\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}}} - \frac{c}{b}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      7. *-commutative 92.1%

         \[\leadsto \left(\frac{-2 \cdot \color{blue}{\left({c}^{3} \cdot {a}^{2}\right)}}{{b}^{5}} - \frac{c}{b}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}} \]

      8. associate-/l* 92.1%

         \[\leadsto \left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]

   6. Simplified 92.1%

      \[\leadsto \color{blue}{\left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{{b}^{3}}{{c}^{2}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 92.1%

         \[\leadsto \left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{\color{blue}{1 \cdot {b}^{3}}}{{c}^{2}}} \]

      2. unpow2 92.1%

         \[\leadsto \left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{1 \cdot {b}^{3}}{\color{blue}{c \cdot c}}} \]

      3. times-frac 92.1%

         \[\leadsto \left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\color{blue}{\frac{1}{c} \cdot \frac{{b}^{3}}{c}}} \]

   8. Applied egg-rr 92.1%

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

      1. associate-*l/ 92.1%

         \[\leadsto \left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\color{blue}{\frac{1 \cdot \frac{{b}^{3}}{c}}{c}}} \]

      2. *-lft-identity 92.1%

         \[\leadsto \left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{\color{blue}{\frac{{b}^{3}}{c}}}{c}} \]

   10. Simplified 92.1%

       \[\leadsto \left(\frac{-2 \cdot \left({c}^{3} \cdot {a}^{2}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\color{blue}{\frac{\frac{{b}^{3}}{c}}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.35:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2 \cdot \left({a}^{2} \cdot {c}^{3}\right)}{{b}^{5}} - \frac{c}{b}\right) - \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}\\ \end{array} \]

Alternative 3: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.35:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0)) -0.35)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (/ 1.0 (- (/ a b) (/ b c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)) <= -0.35) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0)) <= -0.35)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -0.35], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $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)) < -0.34999999999999998

   1. Initial program 84.9%

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

   3. Simplified 85.2%

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

   if -0.34999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in b around inf 86.9%

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

      1. clear-num 86.8%

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

      2. inv-pow 86.8%

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

   6. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{{\left(-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 86.7%

         \[\leadsto \color{blue}{\frac{1}{-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}}} \]

      2. associate-*r/ 86.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot a}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}}} \]

      3. neg-mul-1 86.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{-a}}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      4. associate-/r/ 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{\frac{a}{b} \cdot c} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      5. *-commutative 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{c \cdot \frac{a}{b}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      6. associate-*r/ 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{\frac{c \cdot a}{b}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      7. *-commutative 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\frac{\color{blue}{a \cdot c}}{b} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      8. associate-*r/ 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{a \cdot \frac{c}{b}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      9. fma-def 86.9%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{\mathsf{fma}\left(a, \frac{c}{b}, {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}\right)}}} \]

      10. unpow2 86.9%

          \[\leadsto \frac{1}{\frac{-a}{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{a \cdot c}{{b}^{1.5}} \cdot \frac{a \cdot c}{{b}^{1.5}}}\right)}} \]

      11. times-frac 86.9%

          \[\leadsto \frac{1}{\frac{-a}{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{{b}^{1.5} \cdot {b}^{1.5}}}\right)}} \]

      12. pow-sqr 86.9%

          \[\leadsto \frac{1}{\frac{-a}{\mathsf{fma}\left(a, \frac{c}{b}, \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{\color{blue}{{b}^{\left(2 \cdot 1.5\right)}}}\right)}} \]

      13. metadata-eval 86.9%

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

      14. unpow2 86.9%

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

   8. Simplified 86.9%

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

   9. Taylor expanded in a around 0 87.5%

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

       1. +-commutative 87.5%

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

       2. mul-1-neg 87.5%

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

       3. unsub-neg 87.5%

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

   11. Simplified 87.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.35:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 4: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -0.35:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* (* 4.0 a) c))) b) (* a 2.0))))
   (if (<= t_0 -0.35) t_0 (/ 1.0 (- (/ a b) (/ b c))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.35) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((a / 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) - ((4.0d0 * a) * c))) - b) / (a * 2.0d0)
    if (t_0 <= (-0.35d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / ((a / 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) - ((4.0 * a) * c))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -0.35) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((a / b) - (b / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -0.35:
		tmp = t_0
	else:
		tmp = 1.0 / ((a / b) - (b / c))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -0.35)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - ((4.0 * a) * c))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.35)
		tmp = t_0;
	else
		tmp = 1.0 / ((a / 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.35], t$95$0, N[(1.0 / N[(N[(a / b), $MachinePrecision] - N[(b / c), $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)) < -0.34999999999999998

   1. Initial program 84.9%

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

   if -0.34999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 49.7%

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

      1. *-commutative 49.7%

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

   3. Simplified 49.7%

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

   4. Taylor expanded in b around inf 86.9%

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

      1. clear-num 86.8%

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

      2. inv-pow 86.8%

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

   6. Applied egg-rr 86.7%

      \[\leadsto \color{blue}{{\left(-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 86.7%

         \[\leadsto \color{blue}{\frac{1}{-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}}} \]

      2. associate-*r/ 86.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot a}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}}} \]

      3. neg-mul-1 86.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{-a}}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      4. associate-/r/ 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{\frac{a}{b} \cdot c} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      5. *-commutative 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{c \cdot \frac{a}{b}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      6. associate-*r/ 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{\frac{c \cdot a}{b}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      7. *-commutative 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\frac{\color{blue}{a \cdot c}}{b} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      8. associate-*r/ 86.8%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{a \cdot \frac{c}{b}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

      9. fma-def 86.9%

         \[\leadsto \frac{1}{\frac{-a}{\color{blue}{\mathsf{fma}\left(a, \frac{c}{b}, {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}\right)}}} \]

      10. unpow2 86.9%

          \[\leadsto \frac{1}{\frac{-a}{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{a \cdot c}{{b}^{1.5}} \cdot \frac{a \cdot c}{{b}^{1.5}}}\right)}} \]

      11. times-frac 86.9%

          \[\leadsto \frac{1}{\frac{-a}{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{{b}^{1.5} \cdot {b}^{1.5}}}\right)}} \]

      12. pow-sqr 86.9%

          \[\leadsto \frac{1}{\frac{-a}{\mathsf{fma}\left(a, \frac{c}{b}, \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{\color{blue}{{b}^{\left(2 \cdot 1.5\right)}}}\right)}} \]

      13. metadata-eval 86.9%

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

      14. unpow2 86.9%

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

   8. Simplified 86.9%

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

   9. Taylor expanded in a around 0 87.5%

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

       1. +-commutative 87.5%

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

       2. mul-1-neg 87.5%

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

       3. unsub-neg 87.5%

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

   11. Simplified 87.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2} \leq -0.35:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{b} - \frac{b}{c}}\\ \end{array} \]

Alternative 5: 82.1% 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 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. *-commutative 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in b around inf 80.5%

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

   1. clear-num 80.4%

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

   2. inv-pow 80.4%

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

6. Applied egg-rr 80.4%

   \[\leadsto \color{blue}{{\left(-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 80.4%

      \[\leadsto \color{blue}{\frac{1}{-1 \cdot \frac{a}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}}} \]

   2. associate-*r/ 80.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot a}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}}} \]

   3. neg-mul-1 80.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{-a}}{\frac{a}{\frac{b}{c}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

   4. associate-/r/ 80.4%

      \[\leadsto \frac{1}{\frac{-a}{\color{blue}{\frac{a}{b} \cdot c} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

   5. *-commutative 80.4%

      \[\leadsto \frac{1}{\frac{-a}{\color{blue}{c \cdot \frac{a}{b}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

   6. associate-*r/ 80.4%

      \[\leadsto \frac{1}{\frac{-a}{\color{blue}{\frac{c \cdot a}{b}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

   7. *-commutative 80.4%

      \[\leadsto \frac{1}{\frac{-a}{\frac{\color{blue}{a \cdot c}}{b} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

   8. associate-*r/ 80.4%

      \[\leadsto \frac{1}{\frac{-a}{\color{blue}{a \cdot \frac{c}{b}} + {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}}} \]

   9. fma-def 80.5%

      \[\leadsto \frac{1}{\frac{-a}{\color{blue}{\mathsf{fma}\left(a, \frac{c}{b}, {\left(\frac{a \cdot c}{{b}^{1.5}}\right)}^{2}\right)}}} \]

   10. unpow2 80.5%

       \[\leadsto \frac{1}{\frac{-a}{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{a \cdot c}{{b}^{1.5}} \cdot \frac{a \cdot c}{{b}^{1.5}}}\right)}} \]

   11. times-frac 80.5%

       \[\leadsto \frac{1}{\frac{-a}{\mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{{b}^{1.5} \cdot {b}^{1.5}}}\right)}} \]

   12. pow-sqr 80.5%

       \[\leadsto \frac{1}{\frac{-a}{\mathsf{fma}\left(a, \frac{c}{b}, \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{\color{blue}{{b}^{\left(2 \cdot 1.5\right)}}}\right)}} \]

   13. metadata-eval 80.5%

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

   14. unpow2 80.5%

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

8. Simplified 80.5%

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

9. Taylor expanded in a around 0 81.2%

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

    1. +-commutative 81.2%

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

    2. mul-1-neg 81.2%

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

    3. unsub-neg 81.2%

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

11. Simplified 81.2%

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

12. Final simplification 81.2%

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

Alternative 6: 64.3% 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. *-commutative 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in b around inf 63.6%

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

   1. mul-1-neg 63.6%

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

   2. distribute-neg-frac 63.6%

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

6. Simplified 63.6%

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

7. Final simplification 63.6%

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

Alternative 7: 1.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. *-commutative 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in b around -inf 11.6%

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

   1. +-commutative 11.6%

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

   2. mul-1-neg 11.6%

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

   3. unsub-neg 11.6%

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

6. Simplified 11.6%

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

7. Taylor expanded in c around inf 1.6%

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

8. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2024024 
(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)))