Quadratic roots, narrow range

Percentage Accurate: 55.6% → 91.3%

Time: 11.0s

Alternatives: 9

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: 55.6% 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: 91.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left(-1 + e^{\mathsf{log1p}\left(4 \cdot {\left(c \cdot a\right)}^{4}\right)}\right)}{a \cdot {b}^{7}} - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.2)
   (/ (- (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
       (/
        (+
         (* 16.0 (* (pow a 4.0) (pow c 4.0)))
         (+ -1.0 (exp (log1p (* 4.0 (pow (* c a) 4.0))))))
        (* a (pow b 7.0))))
      (/ (* a (pow c 2.0)) (pow b 3.0)))
     (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.2) {
		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 * (((16.0 * (pow(a, 4.0) * pow(c, 4.0))) + (-1.0 + exp(log1p((4.0 * pow((c * a), 4.0)))))) / (a * pow(b, 7.0)))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.2)
		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(16.0 * Float64((a ^ 4.0) * (c ^ 4.0))) + Float64(-1.0 + exp(log1p(Float64(4.0 * (Float64(c * a) ^ 4.0)))))) / Float64(a * (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.2], 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[(16.0 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + N[Exp[N[Log[1 + N[(4.0 * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $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 b < 3.2000000000000002

   1. Initial program 83.6%

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

      1. sqr-neg 83.6%

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

      2. +-commutative 83.6%

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

      3. unsub-neg 83.6%

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

      4. sqr-neg 83.6%

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

      5. fma-neg 83.7%

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

      6. distribute-lft-neg-in 83.7%

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

      7. *-commutative 83.7%

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

      8. *-commutative 83.7%

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

      9. distribute-rgt-neg-in 83.7%

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

      10. metadata-eval 83.7%

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

      11. *-commutative 83.7%

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

   3. Simplified 83.7%

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

   if 3.2000000000000002 < b

   1. Initial program 45.9%

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

      1. *-commutative 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in b around inf 94.3%

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

      1. expm1-log1p-u 94.3%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

      2. expm1-udef 94.3%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}\right)} - 1\right)}}{a \cdot {b}^{7}}\right)\right) \]

      3. unpow-prod-down 94.3%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{{-2}^{2} \cdot {\left({a}^{2} \cdot {c}^{2}\right)}^{2}}\right)} - 1\right)}{a \cdot {b}^{7}}\right)\right) \]

      4. metadata-eval 94.3%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{4} \cdot {\left({a}^{2} \cdot {c}^{2}\right)}^{2}\right)} - 1\right)}{a \cdot {b}^{7}}\right)\right) \]

      5. pow-prod-down 94.3%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left(e^{\mathsf{log1p}\left(4 \cdot {\color{blue}{\left({\left(a \cdot c\right)}^{2}\right)}}^{2}\right)} - 1\right)}{a \cdot {b}^{7}}\right)\right) \]

      6. pow-pow 94.3%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left(e^{\mathsf{log1p}\left(4 \cdot \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}}\right)} - 1\right)}{a \cdot {b}^{7}}\right)\right) \]

      7. metadata-eval 94.3%

         \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left(e^{\mathsf{log1p}\left(4 \cdot {\left(a \cdot c\right)}^{\color{blue}{4}}\right)} - 1\right)}{a \cdot {b}^{7}}\right)\right) \]

   6. Applied egg-rr 94.3%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(4 \cdot {\left(a \cdot c\right)}^{4}\right)} - 1\right)}}{a \cdot {b}^{7}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left(-1 + e^{\mathsf{log1p}\left(4 \cdot {\left(c \cdot a\right)}^{4}\right)}\right)}{a \cdot {b}^{7}} - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 2: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.2)
   (/ (- (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 (* c a) 4.0) a) (/ 20.0 (pow b 7.0))))
      (/ (* a (pow c 2.0)) (pow b 3.0)))
     (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.2) {
		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((c * a), 4.0) / a) * (20.0 / pow(b, 7.0)))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.2)
		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(c * a) ^ 4.0) / a) * Float64(20.0 / (b ^ 7.0)))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.2], 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[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(20.0 / N[Power[b, 7.0], $MachinePrecision]), $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 b < 3.2000000000000002

   1. Initial program 83.6%

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

      1. sqr-neg 83.6%

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

      2. +-commutative 83.6%

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

      3. unsub-neg 83.6%

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

      4. sqr-neg 83.6%

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

      5. fma-neg 83.7%

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

      6. distribute-lft-neg-in 83.7%

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

      7. *-commutative 83.7%

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

      8. *-commutative 83.7%

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

      9. distribute-rgt-neg-in 83.7%

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

      10. metadata-eval 83.7%

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

      11. *-commutative 83.7%

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

   3. Simplified 83.7%

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

   if 3.2000000000000002 < b

   1. Initial program 45.9%

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

      1. *-commutative 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in b around inf 94.3%

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

   5. Taylor expanded in c around 0 94.3%

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

      1. distribute-rgt-out 94.3%

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

      2. associate-*l* 94.3%

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

      3. *-commutative 94.3%

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

      4. times-frac 94.3%

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

   7. Simplified 94.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{20}{{b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right)\\ \end{array} \]

Alternative 3: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left({c}^{3} \cdot \frac{{a}^{2}}{{b}^{5}}\right) - \frac{c}{b}\right) - {c}^{2} \cdot \frac{a}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (-
    (- (* -2.0 (* (pow c 3.0) (/ (pow a 2.0) (pow b 5.0)))) (/ c b))
    (* (pow c 2.0) (/ a (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((-2.0 * (pow(c, 3.0) * (pow(a, 2.0) / pow(b, 5.0)))) - (c / b)) - (pow(c, 2.0) * (a / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-2.0 * Float64((c ^ 3.0) * Float64((a ^ 2.0) / (b ^ 5.0)))) - Float64(c / b)) - Float64((c ^ 2.0) * Float64(a / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.0], 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[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6

   1. Initial program 83.5%

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

      1. sqr-neg 83.5%

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

      2. +-commutative 83.5%

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

      3. unsub-neg 83.5%

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

      4. sqr-neg 83.5%

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

      5. fma-neg 83.6%

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

      6. distribute-lft-neg-in 83.6%

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

      7. *-commutative 83.6%

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

      8. *-commutative 83.6%

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

      9. distribute-rgt-neg-in 83.6%

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

      10. metadata-eval 83.6%

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

      11. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   if 6 < b

   1. Initial program 45.2%

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

      1. *-commutative 45.2%

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

   3. Simplified 45.2%

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

      1. fma-neg 45.2%

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

      2. *-commutative 45.2%

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

      3. distribute-rgt-neg-in 45.2%

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

      4. distribute-lft-neg-in 45.2%

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

      5. metadata-eval 45.2%

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

      6. *-commutative 45.2%

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

      7. add-cbrt-cube 44.6%

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

      8. pow3 44.7%

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

      9. *-commutative 44.7%

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

      10. associate-*r* 44.7%

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

      11. *-commutative 44.7%

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

   5. Applied egg-rr 44.7%

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

   6. Taylor expanded in b around inf 92.5%

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

      1. associate-+r+ 92.5%

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

      2. mul-1-neg 92.5%

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

      3. unsub-neg 92.5%

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

      4. mul-1-neg 92.5%

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

      5. unsub-neg 92.5%

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

      6. associate-/l* 92.5%

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

      7. associate-/r/ 92.5%

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

      8. associate-/l* 92.5%

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

      9. associate-/r/ 92.5%

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

   8. Simplified 92.5%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left({c}^{3} \cdot \frac{{a}^{2}}{{b}^{5}}\right) - \frac{c}{b}\right) - {c}^{2} \cdot \frac{a}{{b}^{3}}\\ \end{array} \]

Alternative 4: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 240:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{c}^{2}\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 240.0)
   (/ (- (sqrt (fma a (* c -4.0) (* b b))) b) (* a 2.0))
   (- (/ (* a (- (pow c 2.0))) (pow b 3.0)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 240.0) {
		tmp = (sqrt(fma(a, (c * -4.0), (b * b))) - b) / (a * 2.0);
	} else {
		tmp = ((a * -pow(c, 2.0)) / pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 240.0)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -4.0), Float64(b * b))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(-(c ^ 2.0))) / (b ^ 3.0)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 240.0], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * (-N[Power[c, 2.0], $MachinePrecision])), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 240

   1. Initial program 80.3%

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

   3. Simplified 80.3%

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

   if 240 < b

   1. Initial program 41.0%

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

      1. *-commutative 41.0%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in b around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. unsub-neg 91.1%

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

      3. mul-1-neg 91.1%

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

      4. distribute-neg-frac 91.1%

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

   6. Simplified 91.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 240:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{c}^{2}\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 5: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 240:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{c}^{2}\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 240.0)
   (/ (- (sqrt (fma b b (* c (* a -4.0)))) b) (* a 2.0))
   (- (/ (* a (- (pow c 2.0))) (pow b 3.0)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 240.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((a * -pow(c, 2.0)) / pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 240.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(-(c ^ 2.0))) / (b ^ 3.0)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 240.0], 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[(N[(a * (-N[Power[c, 2.0], $MachinePrecision])), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 240

   1. Initial program 80.3%

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

      1. sqr-neg 80.3%

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

      2. +-commutative 80.3%

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

      3. unsub-neg 80.3%

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

      4. sqr-neg 80.3%

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

      5. fma-neg 80.4%

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

      6. distribute-lft-neg-in 80.4%

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

      7. *-commutative 80.4%

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

      8. *-commutative 80.4%

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

      9. distribute-rgt-neg-in 80.4%

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

      10. metadata-eval 80.4%

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

      11. *-commutative 80.4%

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

   3. Simplified 80.4%

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

   if 240 < b

   1. Initial program 41.0%

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

      1. *-commutative 41.0%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in b around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. unsub-neg 91.1%

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

      3. mul-1-neg 91.1%

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

      4. distribute-neg-frac 91.1%

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

   6. Simplified 91.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 240:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{c}^{2}\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 6: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 240:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{c}^{2}\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 240.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (- (/ (* a (- (pow c 2.0))) (pow b 3.0)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 240.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((a * -pow(c, 2.0)) / pow(b, 3.0)) - (c / b);
	}
	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 <= 240.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = ((a * -(c ** 2.0d0)) / (b ** 3.0d0)) - (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 240.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = ((a * -Math.pow(c, 2.0)) / Math.pow(b, 3.0)) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 240.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = ((a * -math.pow(c, 2.0)) / math.pow(b, 3.0)) - (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 240.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(a * Float64(-(c ^ 2.0))) / (b ^ 3.0)) - Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 240.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = ((a * -(c ^ 2.0)) / (b ^ 3.0)) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 240.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * (-N[Power[c, 2.0], $MachinePrecision])), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 240

   1. Initial program 80.3%

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

   if 240 < b

   1. Initial program 41.0%

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

      1. *-commutative 41.0%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in b around inf 91.1%

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

      1. mul-1-neg 91.1%

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

      2. unsub-neg 91.1%

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

      3. mul-1-neg 91.1%

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

      4. distribute-neg-frac 91.1%

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

   6. Simplified 91.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 240:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-{c}^{2}\right)}{{b}^{3}} - \frac{c}{b}\\ \end{array} \]

Alternative 7: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3800:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3800.0)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3800.0) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	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 <= 3800.0d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3800.0) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3800.0:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3800.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3800.0)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3800.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3800

   1. Initial program 76.0%

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

   if 3800 < b

   1. Initial program 36.1%

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

      1. *-commutative 36.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in b around inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. distribute-neg-frac 80.2%

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

   6. Simplified 80.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3800:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8: 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 53.7%

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

   1. *-commutative 53.7%

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

3. Simplified 53.7%

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

4. Taylor expanded in b around inf 65.3%

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

   1. mul-1-neg 65.3%

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

   2. distribute-neg-frac 65.3%

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

6. Simplified 65.3%

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

7. Final simplification 65.3%

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

Alternative 9: 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 53.7%

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

   1. *-commutative 53.7%

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

3. Simplified 53.7%

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

4. Taylor expanded in b around -inf 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 2023322 
(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)))