Quadratic roots, narrow range

Percentage Accurate: 55.2% → 91.9%

Time: 16.4s

Alternatives: 10

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: 55.2% 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: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0066:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0066)
   (/ (- (sqrt (* c (+ (* -4.0 a) (/ (pow b 2.0) c)))) b) (* a 2.0))
   (-
    (*
     a
     (-
      (*
       a
       (+
        (* -2.0 (/ (pow c 3.0) (pow b 5.0)))
        (* -0.25 (/ (* a (/ (* (pow c 4.0) 20.0) (pow b 6.0))) b))))
      (/ (pow c 2.0) (pow b 3.0))))
    (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0066) {
		tmp = (sqrt((c * ((-4.0 * a) + (pow(b, 2.0) / c)))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((a * ((-2.0 * (pow(c, 3.0) / pow(b, 5.0))) + (-0.25 * ((a * ((pow(c, 4.0) * 20.0) / pow(b, 6.0))) / b)))) - (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 <= 0.0066d0) then
        tmp = (sqrt((c * (((-4.0d0) * a) + ((b ** 2.0d0) / c)))) - b) / (a * 2.0d0)
    else
        tmp = (a * ((a * (((-2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0))) + ((-0.25d0) * ((a * (((c ** 4.0d0) * 20.0d0) / (b ** 6.0d0))) / b)))) - ((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 <= 0.0066) {
		tmp = (Math.sqrt((c * ((-4.0 * a) + (Math.pow(b, 2.0) / c)))) - b) / (a * 2.0);
	} else {
		tmp = (a * ((a * ((-2.0 * (Math.pow(c, 3.0) / Math.pow(b, 5.0))) + (-0.25 * ((a * ((Math.pow(c, 4.0) * 20.0) / Math.pow(b, 6.0))) / b)))) - (Math.pow(c, 2.0) / Math.pow(b, 3.0)))) - (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.0066:
		tmp = (math.sqrt((c * ((-4.0 * a) + (math.pow(b, 2.0) / c)))) - b) / (a * 2.0)
	else:
		tmp = (a * ((a * ((-2.0 * (math.pow(c, 3.0) / math.pow(b, 5.0))) + (-0.25 * ((a * ((math.pow(c, 4.0) * 20.0) / math.pow(b, 6.0))) / b)))) - (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 <= 0.0066)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(-4.0 * a) + Float64((b ^ 2.0) / c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(a * Float64(Float64(a * Float64(Float64(-2.0 * Float64((c ^ 3.0) / (b ^ 5.0))) + Float64(-0.25 * Float64(Float64(a * Float64(Float64((c ^ 4.0) * 20.0) / (b ^ 6.0))) / b)))) - 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 <= 0.0066)
		tmp = (sqrt((c * ((-4.0 * a) + ((b ^ 2.0) / c)))) - b) / (a * 2.0);
	else
		tmp = (a * ((a * ((-2.0 * ((c ^ 3.0) / (b ^ 5.0))) + (-0.25 * ((a * (((c ^ 4.0) * 20.0) / (b ^ 6.0))) / b)))) - ((c ^ 2.0) / (b ^ 3.0)))) - (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0066], N[(N[(N[Sqrt[N[(c * N[(N[(-4.0 * a), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(a * N[(N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(a * N[(N[(N[Power[c, 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0066

   1. Initial program 87.4%

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

      1. *-commutative 87.4%

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

      2. +-commutative 87.4%

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

      3. sqr-neg 87.4%

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

      4. unsub-neg 87.4%

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

      5. sqr-neg 87.4%

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

      6. fma-neg 87.5%

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

      7. distribute-lft-neg-in 87.5%

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

      8. *-commutative 87.5%

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

      9. *-commutative 87.5%

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

      10. distribute-rgt-neg-in 87.5%

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

      11. metadata-eval 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in c around inf 87.6%

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

   if 0.0066 < b

   1. Initial program 50.8%

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

      1. *-commutative 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in a around 0 94.3%

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

   6. Taylor expanded in b around 0 94.3%

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

      1. distribute-rgt-out 94.3%

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

      2. metadata-eval 94.3%

         \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \frac{{c}^{4} \cdot \color{blue}{20}}{{b}^{6}}}{b}\right)\right) \]

   8. Simplified 94.3%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \color{blue}{\frac{{c}^{4} \cdot 20}{{b}^{6}}}}{b}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0066:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-4 \cdot a + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \frac{{c}^{4} \cdot 20}{{b}^{6}}}{b}\right) - \frac{{c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.21:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 \cdot {a}^{2}\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.21)
   (/ (- (sqrt (fma b b (* c (* -4.0 a)))) b) (* a 2.0))
   (/
    (-
     (- (* (* -2.0 (pow a 2.0)) (/ (pow c 3.0) (pow b 4.0))) c)
     (/ (* a (pow c 2.0)) (pow b 2.0)))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.21) {
		tmp = (sqrt(fma(b, b, (c * (-4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = ((((-2.0 * pow(a, 2.0)) * (pow(c, 3.0) / pow(b, 4.0))) - c) - ((a * pow(c, 2.0)) / pow(b, 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.21)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-2.0 * (a ^ 2.0)) * Float64((c ^ 3.0) / (b ^ 4.0))) - c) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.21], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.209999999999999992

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

      2. +-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. unsub-neg 82.5%

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

      5. sqr-neg 82.5%

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

      6. fma-neg 82.7%

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

      7. distribute-lft-neg-in 82.7%

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

      8. *-commutative 82.7%

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

      9. *-commutative 82.7%

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

      10. distribute-rgt-neg-in 82.7%

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

      11. metadata-eval 82.7%

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

   3. Simplified 82.7%

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

   if 0.209999999999999992 < b

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in b around inf 93.0%

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

      1. associate-+r+ 93.0%

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

      2. mul-1-neg 93.0%

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

      3. unsub-neg 93.0%

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

      4. mul-1-neg 93.0%

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

      5. unsub-neg 93.0%

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

      6. associate-/l* 93.0%

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

      7. associate-*r* 93.0%

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

   7. Simplified 93.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.21:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-2 \cdot {a}^{2}\right) \cdot \frac{{c}^{3}}{{b}^{4}} - c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.215:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right) + 2 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.215)
   (/ (- (sqrt (fma b b (* c (* -4.0 a)))) b) (* a 2.0))
   (/
    (+
     (fma a (pow (/ c (- b)) 2.0) c)
     (* 2.0 (* (pow a 2.0) (/ (pow c 3.0) (pow b 4.0)))))
    (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.215) {
		tmp = (sqrt(fma(b, b, (c * (-4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = (fma(a, pow((c / -b), 2.0), c) + (2.0 * (pow(a, 2.0) * (pow(c, 3.0) / pow(b, 4.0))))) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.215)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(fma(a, (Float64(c / Float64(-b)) ^ 2.0), c) + Float64(2.0 * Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 4.0))))) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.215], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] + N[(2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.214999999999999997

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

      2. +-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. unsub-neg 82.5%

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

      5. sqr-neg 82.5%

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

      6. fma-neg 82.7%

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

      7. distribute-lft-neg-in 82.7%

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

      8. *-commutative 82.7%

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

      9. *-commutative 82.7%

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

      10. distribute-rgt-neg-in 82.7%

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

      11. metadata-eval 82.7%

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

   3. Simplified 82.7%

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

   if 0.214999999999999997 < b

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in c around 0 92.8%

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

   6. Taylor expanded in b around -inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. distribute-neg-frac2 93.0%

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

   8. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, {\left(\frac{-c}{b}\right)}^{2}, c\right) + 2 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}\right)}{-b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.215:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right) + 2 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{4}}\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.195:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \left(\frac{{c}^{2}}{{b}^{3}} + 2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.195)
   (/ (- (sqrt (fma b b (* c (* -4.0 a)))) b) (* a 2.0))
   (-
    (/ c (- b))
    (*
     a
     (+
      (/ (pow c 2.0) (pow b 3.0))
      (* 2.0 (/ (* a (pow c 3.0)) (pow b 5.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.195) {
		tmp = (sqrt(fma(b, b, (c * (-4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = (c / -b) - (a * ((pow(c, 2.0) / pow(b, 3.0)) + (2.0 * ((a * pow(c, 3.0)) / pow(b, 5.0)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.195)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64(Float64((c ^ 2.0) / (b ^ 3.0)) + Float64(2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.195], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.19500000000000001

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

      2. +-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. unsub-neg 82.5%

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

      5. sqr-neg 82.5%

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

      6. fma-neg 82.7%

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

      7. distribute-lft-neg-in 82.7%

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

      8. *-commutative 82.7%

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

      9. *-commutative 82.7%

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

      10. distribute-rgt-neg-in 82.7%

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

      11. metadata-eval 82.7%

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

   3. Simplified 82.7%

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

   if 0.19500000000000001 < b

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in c around 0 92.8%

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

   6. Taylor expanded in c around -inf 92.6%

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

   7. Taylor expanded in a around 0 93.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.195:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \left(\frac{{c}^{2}}{{b}^{3}} + 2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-2 \cdot {\left(c \cdot a\right)}^{2} - c \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}} + \frac{-1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.2)
   (/ (- (sqrt (fma b b (* c (* -4.0 a)))) b) (* a 2.0))
   (*
    c
    (+
     (/ (- (* -2.0 (pow (* c a) 2.0)) (* c (* a (pow b 2.0)))) (pow b 5.0))
     (/ -1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.2) {
		tmp = (sqrt(fma(b, b, (c * (-4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((((-2.0 * pow((c * a), 2.0)) - (c * (a * pow(b, 2.0)))) / pow(b, 5.0)) + (-1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(Float64(Float64(-2.0 * (Float64(c * a) ^ 2.0)) - Float64(c * Float64(a * (b ^ 2.0)))) / (b ^ 5.0)) + Float64(-1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.2], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(N[(N[(-2.0 * N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(c * N[(a * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.20000000000000001

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

      2. +-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. unsub-neg 82.5%

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

      5. sqr-neg 82.5%

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

      6. fma-neg 82.7%

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

      7. distribute-lft-neg-in 82.7%

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

      8. *-commutative 82.7%

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

      9. *-commutative 82.7%

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

      10. distribute-rgt-neg-in 82.7%

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

      11. metadata-eval 82.7%

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

   3. Simplified 82.7%

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

   if 0.20000000000000001 < b

   1. Initial program 48.4%

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

      1. *-commutative 48.4%

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

   3. Simplified 48.4%

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

   5. Taylor expanded in c around 0 92.8%

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

   6. Taylor expanded in b around 0 92.8%

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

      1. mul-1-neg 92.8%

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

      2. unsub-neg 92.8%

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

      3. *-commutative 92.8%

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

      4. unpow2 92.8%

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

      5. unpow2 92.8%

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

      6. swap-sqr 92.8%

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

      7. unpow2 92.8%

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

      8. associate-*r* 92.8%

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

   8. Simplified 92.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-2 \cdot {\left(c \cdot a\right)}^{2} - c \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}} + \frac{-1}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.92:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.92)
   (/ (- (sqrt (fma b b (* c (* -4.0 a)))) b) (* a 2.0))
   (/ (fma a (pow (/ c (- b)) 2.0) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.92) {
		tmp = (sqrt(fma(b, b, (c * (-4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = fma(a, pow((c / -b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.92)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(-4.0 * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, (Float64(c / Float64(-b)) ^ 2.0), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.92], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9199999999999999

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. +-commutative 81.4%

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

      3. sqr-neg 81.4%

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

      4. unsub-neg 81.4%

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

      5. sqr-neg 81.4%

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

      6. fma-neg 81.5%

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

      7. distribute-lft-neg-in 81.5%

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

      8. *-commutative 81.5%

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

      9. *-commutative 81.5%

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

      10. distribute-rgt-neg-in 81.5%

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

      11. metadata-eval 81.5%

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

   3. Simplified 81.5%

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

   if 1.9199999999999999 < b

   1. Initial program 46.9%

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

      1. *-commutative 46.9%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in a around 0 88.7%

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

      1. mul-1-neg 88.7%

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

      2. unsub-neg 88.7%

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

      3. mul-1-neg 88.7%

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

      4. distribute-neg-frac2 88.7%

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

      5. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

   8. Taylor expanded in b around inf 88.7%

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

      1. distribute-lft-out 88.7%

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

      2. associate-*r/ 88.7%

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

      3. mul-1-neg 88.7%

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

      4. distribute-neg-frac2 88.7%

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

      5. +-commutative 88.7%

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

      6. associate-/l* 88.7%

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

      7. fma-define 88.7%

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

      8. unpow2 88.7%

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

      9. unpow2 88.7%

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

      10. times-frac 88.7%

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

      11. sqr-neg 88.7%

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

      12. distribute-frac-neg2 88.7%

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

      13. distribute-frac-neg2 88.7%

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

      14. unpow2 88.7%

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

      15. distribute-frac-neg2 88.7%

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

      16. distribute-frac-neg 88.7%

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

   10. Simplified 88.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.92:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.92:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.92)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (/ (fma a (pow (/ c (- b)) 2.0) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.92) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = fma(a, pow((c / -b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.92)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(fma(a, (Float64(c / Float64(-b)) ^ 2.0), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.92], 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[(a * N[Power[N[(c / (-b)), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9199999999999999

   1. Initial program 81.4%

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

   if 1.9199999999999999 < b

   1. Initial program 46.9%

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

      1. *-commutative 46.9%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in a around 0 88.7%

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

      1. mul-1-neg 88.7%

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

      2. unsub-neg 88.7%

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

      3. mul-1-neg 88.7%

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

      4. distribute-neg-frac2 88.7%

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

      5. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

   8. Taylor expanded in b around inf 88.7%

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

      1. distribute-lft-out 88.7%

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

      2. associate-*r/ 88.7%

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

      3. mul-1-neg 88.7%

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

      4. distribute-neg-frac2 88.7%

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

      5. +-commutative 88.7%

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

      6. associate-/l* 88.7%

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

      7. fma-define 88.7%

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

      8. unpow2 88.7%

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

      9. unpow2 88.7%

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

      10. times-frac 88.7%

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

      11. sqr-neg 88.7%

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

      12. distribute-frac-neg2 88.7%

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

      13. distribute-frac-neg2 88.7%

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

      14. unpow2 88.7%

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

      15. distribute-frac-neg2 88.7%

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

      16. distribute-frac-neg 88.7%

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

   10. Simplified 88.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.92:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{-b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - c \cdot \frac{a}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.9)
   (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
   (* c (- (/ -1.0 b) (* c (/ a (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.9) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - (c * (a / pow(b, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.9d0) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c * (((-1.0d0) / b) - (c * (a / (b ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.9) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c * ((-1.0 / b) - (c * (a / Math.pow(b, 3.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.9:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = c * ((-1.0 / b) - (c * (a / math.pow(b, 3.0))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.9)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(c * Float64(a / (b ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.9)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = c * ((-1.0 / b) - (c * (a / (b ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.9], 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 * N[(N[(-1.0 / b), $MachinePrecision] - N[(c * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8999999999999999

   1. Initial program 81.4%

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

   if 1.8999999999999999 < b

   1. Initial program 46.9%

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

      1. *-commutative 46.9%

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

   3. Simplified 46.9%

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

   5. Taylor expanded in a around 0 88.7%

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

      1. mul-1-neg 88.7%

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

      2. unsub-neg 88.7%

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

      3. mul-1-neg 88.7%

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

      4. distribute-neg-frac2 88.7%

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

      5. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

      1. div-inv 88.5%

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

      2. fma-neg 88.5%

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

      3. *-commutative 88.5%

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

      4. div-inv 88.5%

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

      5. pow-flip 88.5%

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

      6. metadata-eval 88.5%

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

   9. Applied egg-rr 88.5%

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

       1. fma-undefine 88.5%

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

       2. unsub-neg 88.5%

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

       3. distribute-frac-neg2 88.5%

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

       4. distribute-neg-frac 88.5%

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

       5. metadata-eval 88.5%

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

       6. associate-*l* 88.5%

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

   11. Simplified 88.5%

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

   12. Taylor expanded in c around 0 88.5%

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

       1. sub-neg 88.5%

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

       2. distribute-neg-frac 88.5%

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

       3. metadata-eval 88.5%

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

       4. +-commutative 88.5%

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

       5. mul-1-neg 88.5%

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

       6. unsub-neg 88.5%

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

       7. *-commutative 88.5%

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

       8. associate-*r/ 88.5%

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

   14. Simplified 88.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - c \cdot \frac{a}{{b}^{3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - c \cdot \frac{a}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (* c (/ a (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (c * (a / pow(b, 3.0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (((-1.0d0) / b) - (c * (a / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - (c * (a / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - (c * (a / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(c * Float64(a / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - (c * (a / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(c * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

5. Taylor expanded in a around 0 83.0%

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

   1. mul-1-neg 83.0%

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

   2. unsub-neg 83.0%

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

   3. mul-1-neg 83.0%

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

   4. distribute-neg-frac2 83.0%

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

   5. associate-/l* 83.0%

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

7. Simplified 83.0%

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

   1. div-inv 82.8%

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

   2. fma-neg 82.9%

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

   3. *-commutative 82.9%

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

   4. div-inv 82.9%

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

   5. pow-flip 82.9%

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

   6. metadata-eval 82.9%

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

9. Applied egg-rr 82.9%

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

    1. fma-undefine 82.8%

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

    2. unsub-neg 82.8%

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

    3. distribute-frac-neg2 82.8%

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

    4. distribute-neg-frac 82.8%

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

    5. metadata-eval 82.8%

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

    6. associate-*l* 82.8%

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

11. Simplified 82.8%

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

12. Taylor expanded in c around 0 82.9%

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

    1. sub-neg 82.9%

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

    2. distribute-neg-frac 82.9%

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

    3. metadata-eval 82.9%

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

    4. +-commutative 82.9%

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

    5. mul-1-neg 82.9%

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

    6. unsub-neg 82.9%

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

    7. *-commutative 82.9%

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

    8. associate-*r/ 82.9%

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

14. Simplified 82.9%

    \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{b} - c \cdot \frac{a}{{b}^{3}}\right)} \]
15. Add Preprocessing

Alternative 10: 64.6% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Add Preprocessing

5. Taylor expanded in b around inf 65.6%

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

   1. associate-*r/ 65.6%

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

   2. mul-1-neg 65.6%

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

7. Simplified 65.6%

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

8. Final simplification 65.6%

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

Reproduce

herbie shell --seed 2024106 
(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)))