Cubic critical

Percentage Accurate: 51.8% → 84.4%

Time: 15.4s

Alternatives: 12

Speedup: 16.4×

Specification

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

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

Wolfram

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

Initial Program: 51.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

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

Wolfram

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

Alternative 1: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -15500000:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-77}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -15500000.0)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 4e-77)
     (* (- b (hypot b (sqrt (* c (* a -3.0))))) (/ -0.3333333333333333 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -15500000.0) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 4e-77) {
		tmp = (b - hypot(b, sqrt((c * (a * -3.0))))) * (-0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -15500000.0) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 4e-77) {
		tmp = (b - Math.hypot(b, Math.sqrt((c * (a * -3.0))))) * (-0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -15500000.0:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 4e-77:
		tmp = (b - math.hypot(b, math.sqrt((c * (a * -3.0))))) * (-0.3333333333333333 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -15500000.0)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 4e-77)
		tmp = Float64(Float64(b - hypot(b, sqrt(Float64(c * Float64(a * -3.0))))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -15500000.0)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 4e-77)
		tmp = (b - hypot(b, sqrt((c * (a * -3.0))))) * (-0.3333333333333333 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -15500000.0], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-77], N[(N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.55e7

   1. Initial program 66.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 97.4%

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

      1. *-commutative 97.4%

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

   4. Simplified 97.4%

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

   if -1.55e7 < b < 3.9999999999999997e-77

   1. Initial program 75.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. /-rgt-identity 75.6%

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

      2. metadata-eval 75.6%

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

      3. associate-/r/ 75.6%

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

      4. metadata-eval 75.6%

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

      5. metadata-eval 75.6%

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

      6. times-frac 75.6%

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

      7. *-commutative 75.6%

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

      8. times-frac 75.5%

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

      9. *-commutative 75.5%

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

      10. associate-/r* 75.4%

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

      11. associate-*l/ 75.4%

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

   3. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 50.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}\right)\right)} \]

      2. expm1-udef 27.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}\right)} - 1} \]

   5. Applied egg-rr 28.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 52.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}\right)\right)} \]

      2. expm1-log1p 77.9%

         \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}} \]

      3. associate-*r/ 77.9%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}{a}} \]

      4. *-commutative 77.9%

         \[\leadsto \frac{\color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot -0.3333333333333333}}{a} \]

      5. associate-*r/ 78.0%

         \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot \frac{-0.3333333333333333}{a}} \]

      6. /-rgt-identity 78.0%

         \[\leadsto \color{blue}{\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{1}} \cdot \frac{-0.3333333333333333}{a} \]

      7. /-rgt-identity 78.0%

         \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)} \cdot \frac{-0.3333333333333333}{a} \]

      8. associate-*r* 77.9%

         \[\leadsto \left(b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}\right)\right) \cdot \frac{-0.3333333333333333}{a} \]

      9. *-commutative 77.9%

         \[\leadsto \left(b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3}\right)\right) \cdot \frac{-0.3333333333333333}{a} \]

      10. associate-*l* 77.9%

          \[\leadsto \left(b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}\right)\right) \cdot \frac{-0.3333333333333333}{a} \]

   7. Simplified 77.9%

      \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{-0.3333333333333333}{a}} \]

   if 3.9999999999999997e-77 < b

   1. Initial program 17.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 87.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -15500000:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-77}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+135}:\\ \;\;\;\;\left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \frac{1}{\frac{a}{0.3333333333333333}}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e+135)
   (* (- (- (* (/ c b) (* a 1.5)) b) b) (/ 1.0 (/ a 0.3333333333333333)))
   (if (<= b 2.5e-77)
     (* (/ 0.3333333333333333 a) (- (sqrt (- (* b b) (* 3.0 (* a c)))) b))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e+135) {
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (1.0 / (a / 0.3333333333333333));
	} else if (b <= 2.5e-77) {
		tmp = (0.3333333333333333 / a) * (sqrt(((b * b) - (3.0 * (a * c)))) - b);
	} else {
		tmp = -0.5 * (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 <= (-7.5d+135)) then
        tmp = ((((c / b) * (a * 1.5d0)) - b) - b) * (1.0d0 / (a / 0.3333333333333333d0))
    else if (b <= 2.5d-77) then
        tmp = (0.3333333333333333d0 / a) * (sqrt(((b * b) - (3.0d0 * (a * c)))) - b)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e+135) {
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (1.0 / (a / 0.3333333333333333));
	} else if (b <= 2.5e-77) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e+135:
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (1.0 / (a / 0.3333333333333333))
	elif b <= 2.5e-77:
		tmp = (0.3333333333333333 / a) * (math.sqrt(((b * b) - (3.0 * (a * c)))) - b)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e+135)
		tmp = Float64(Float64(Float64(Float64(Float64(c / b) * Float64(a * 1.5)) - b) - b) * Float64(1.0 / Float64(a / 0.3333333333333333)));
	elseif (b <= 2.5e-77)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e+135)
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (1.0 / (a / 0.3333333333333333));
	elseif (b <= 2.5e-77)
		tmp = (0.3333333333333333 / a) * (sqrt(((b * b) - (3.0 * (a * c)))) - b);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e+135], N[(N[(N[(N[(N[(c / b), $MachinePrecision] * N[(a * 1.5), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] * N[(1.0 / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.5e-77], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.49999999999999947e135

   1. Initial program 41.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 41.2%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. associate-+l- 41.2%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      3. sub0-neg 41.2%

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

      4. neg-mul-1 41.2%

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

      5. associate-*r/ 41.2%

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

      6. *-commutative 41.2%

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

      7. metadata-eval 41.2%

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

      8. metadata-eval 41.2%

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

      9. times-frac 41.2%

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

      10. *-commutative 41.2%

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

      11. times-frac 41.2%

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

   3. Simplified 41.2%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 41.2%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -3\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      2. associate-*r* 41.2%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot -3}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      3. *-commutative 41.2%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      4. metadata-eval 41.2%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      5. cancel-sign-sub-inv 41.2%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   5. Applied egg-rr 41.2%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   6. Taylor expanded in b around -inf 84.8%

      \[\leadsto \left(\color{blue}{\left(1.5 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]
   7. Step-by-step derivation

      1. neg-mul-1 84.8%

         \[\leadsto \left(\left(1.5 \cdot \frac{c \cdot a}{b} + \color{blue}{\left(-b\right)}\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

      2. unsub-neg 84.8%

         \[\leadsto \left(\color{blue}{\left(1.5 \cdot \frac{c \cdot a}{b} - b\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      3. associate-*l/ 99.6%

         \[\leadsto \left(\left(1.5 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)} - b\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

      4. *-commutative 99.6%

         \[\leadsto \left(\left(\color{blue}{\left(\frac{c}{b} \cdot a\right) \cdot 1.5} - b\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

      5. associate-*l* 99.6%

         \[\leadsto \left(\left(\color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)} - b\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

   8. Simplified 99.6%

      \[\leadsto \left(\color{blue}{\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]
   9. Step-by-step derivation

      1. clear-num 99.7%

         \[\leadsto \left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \color{blue}{\frac{1}{\frac{a}{0.3333333333333333}}} \]

      2. inv-pow 99.7%

         \[\leadsto \left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]

   10. Applied egg-rr 99.7%

       \[\leadsto \left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \color{blue}{{\left(\frac{a}{0.3333333333333333}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 99.7%

          \[\leadsto \left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \color{blue}{\frac{1}{\frac{a}{0.3333333333333333}}} \]

   12. Simplified 99.7%

       \[\leadsto \left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \color{blue}{\frac{1}{\frac{a}{0.3333333333333333}}} \]

   if -7.49999999999999947e135 < b < 2.49999999999999982e-77

   1. Initial program 82.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 82.1%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. associate-+l- 82.1%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      3. sub0-neg 82.1%

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

      4. neg-mul-1 82.1%

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

      5. associate-*r/ 82.1%

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

      6. *-commutative 82.1%

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

      7. metadata-eval 82.1%

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

      8. metadata-eval 82.1%

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

      9. times-frac 82.1%

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

      10. *-commutative 82.1%

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

      11. times-frac 82.0%

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

   3. Simplified 82.0%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 82.0%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -3\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      2. associate-*r* 81.9%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot -3}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      3. *-commutative 81.9%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      4. metadata-eval 81.9%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      5. cancel-sign-sub-inv 81.9%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   5. Applied egg-rr 81.9%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   if 2.49999999999999982e-77 < b

   1. Initial program 17.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 87.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+135}:\\ \;\;\;\;\left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \frac{1}{\frac{a}{0.3333333333333333}}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;\left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e+150)
   (* (- (- (* (/ c b) (* a 1.5)) b) b) (/ 0.3333333333333333 a))
   (if (<= b 2.9e-77)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e+150) {
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (0.3333333333333333 / a);
	} else if (b <= 2.9e-77) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (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 <= (-4.6d+150)) then
        tmp = ((((c / b) * (a * 1.5d0)) - b) - b) * (0.3333333333333333d0 / a)
    else if (b <= 2.9d-77) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e+150) {
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (0.3333333333333333 / a);
	} else if (b <= 2.9e-77) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.6e+150:
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (0.3333333333333333 / a)
	elif b <= 2.9e-77:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e+150)
		tmp = Float64(Float64(Float64(Float64(Float64(c / b) * Float64(a * 1.5)) - b) - b) * Float64(0.3333333333333333 / a));
	elseif (b <= 2.9e-77)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.6e+150)
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (0.3333333333333333 / a);
	elseif (b <= 2.9e-77)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e+150], N[(N[(N[(N[(N[(c / b), $MachinePrecision] * N[(a * 1.5), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e-77], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.60000000000000002e150

   1. Initial program 34.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 34.1%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. associate-+l- 34.1%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      3. sub0-neg 34.1%

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

      4. neg-mul-1 34.1%

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

      5. associate-*r/ 34.1%

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

      6. *-commutative 34.1%

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

      7. metadata-eval 34.1%

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

      8. metadata-eval 34.1%

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

      9. times-frac 34.1%

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

      10. *-commutative 34.1%

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

      11. times-frac 34.1%

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

   3. Simplified 34.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 34.1%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -3\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      2. associate-*r* 34.1%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot -3}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      3. *-commutative 34.1%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      4. metadata-eval 34.1%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      5. cancel-sign-sub-inv 34.1%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   5. Applied egg-rr 34.1%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   6. Taylor expanded in b around -inf 82.9%

      \[\leadsto \left(\color{blue}{\left(1.5 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]
   7. Step-by-step derivation

      1. neg-mul-1 82.9%

         \[\leadsto \left(\left(1.5 \cdot \frac{c \cdot a}{b} + \color{blue}{\left(-b\right)}\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

      2. unsub-neg 82.9%

         \[\leadsto \left(\color{blue}{\left(1.5 \cdot \frac{c \cdot a}{b} - b\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      3. associate-*l/ 99.6%

         \[\leadsto \left(\left(1.5 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)} - b\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

      4. *-commutative 99.6%

         \[\leadsto \left(\left(\color{blue}{\left(\frac{c}{b} \cdot a\right) \cdot 1.5} - b\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

      5. associate-*l* 99.6%

         \[\leadsto \left(\left(\color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)} - b\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

   8. Simplified 99.6%

      \[\leadsto \left(\color{blue}{\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   if -4.60000000000000002e150 < b < 2.8999999999999999e-77

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 82.8%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. associate-+l- 82.8%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      3. sub0-neg 82.8%

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

      4. neg-mul-1 82.8%

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

      5. associate-*r/ 82.8%

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

      6. metadata-eval 82.8%

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

      7. metadata-eval 82.8%

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

      8. times-frac 82.8%

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

      9. *-commutative 82.8%

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

      10. times-frac 82.6%

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

      11. associate-*l/ 82.8%

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

   3. Simplified 82.7%

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

   if 2.8999999999999999e-77 < b

   1. Initial program 17.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 87.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;\left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+151}:\\ \;\;\;\;\left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e+151)
   (* (- (- (* (/ c b) (* a 1.5)) b) b) (/ 0.3333333333333333 a))
   (if (<= b 4.5e-78)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.12e+151) {
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (0.3333333333333333 / a);
	} else if (b <= 4.5e-78) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (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 <= (-1.12d+151)) then
        tmp = ((((c / b) * (a * 1.5d0)) - b) - b) * (0.3333333333333333d0 / a)
    else if (b <= 4.5d-78) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.12e+151) {
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (0.3333333333333333 / a);
	} else if (b <= 4.5e-78) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.12e+151:
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (0.3333333333333333 / a)
	elif b <= 4.5e-78:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.12e+151)
		tmp = Float64(Float64(Float64(Float64(Float64(c / b) * Float64(a * 1.5)) - b) - b) * Float64(0.3333333333333333 / a));
	elseif (b <= 4.5e-78)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.12e+151)
		tmp = ((((c / b) * (a * 1.5)) - b) - b) * (0.3333333333333333 / a);
	elseif (b <= 4.5e-78)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e+151], N[(N[(N[(N[(N[(c / b), $MachinePrecision] * N[(a * 1.5), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-78], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.12000000000000004e151

   1. Initial program 34.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. neg-sub0 34.1%

         \[\leadsto \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

      2. associate-+l- 34.1%

         \[\leadsto \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]

      3. sub0-neg 34.1%

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

      4. neg-mul-1 34.1%

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

      5. associate-*r/ 34.1%

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

      6. *-commutative 34.1%

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

      7. metadata-eval 34.1%

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

      8. metadata-eval 34.1%

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

      9. times-frac 34.1%

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

      10. *-commutative 34.1%

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

      11. times-frac 34.1%

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

   3. Simplified 34.1%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
   4. Step-by-step derivation

      1. fma-udef 34.1%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -3\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      2. associate-*r* 34.1%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot -3}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      3. *-commutative 34.1%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{-3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      4. metadata-eval 34.1%

         \[\leadsto \left(\sqrt{b \cdot b + \color{blue}{\left(-3\right)} \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      5. cancel-sign-sub-inv 34.1%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   5. Applied egg-rr 34.1%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b - 3 \cdot \left(a \cdot c\right)}} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   6. Taylor expanded in b around -inf 82.9%

      \[\leadsto \left(\color{blue}{\left(1.5 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]
   7. Step-by-step derivation

      1. neg-mul-1 82.9%

         \[\leadsto \left(\left(1.5 \cdot \frac{c \cdot a}{b} + \color{blue}{\left(-b\right)}\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

      2. unsub-neg 82.9%

         \[\leadsto \left(\color{blue}{\left(1.5 \cdot \frac{c \cdot a}{b} - b\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]

      3. associate-*l/ 99.6%

         \[\leadsto \left(\left(1.5 \cdot \color{blue}{\left(\frac{c}{b} \cdot a\right)} - b\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

      4. *-commutative 99.6%

         \[\leadsto \left(\left(\color{blue}{\left(\frac{c}{b} \cdot a\right) \cdot 1.5} - b\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

      5. associate-*l* 99.6%

         \[\leadsto \left(\left(\color{blue}{\frac{c}{b} \cdot \left(a \cdot 1.5\right)} - b\right) - b\right) \cdot \frac{0.3333333333333333}{a} \]

   8. Simplified 99.6%

      \[\leadsto \left(\color{blue}{\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]

   if -1.12000000000000004e151 < b < 4.5e-78

   1. Initial program 82.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   if 4.5e-78 < b

   1. Initial program 17.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 87.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+151}:\\ \;\;\;\;\left(\left(\frac{c}{b} \cdot \left(a \cdot 1.5\right) - b\right) - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.66 \cdot 10^{-195}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.66e-195)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 4.8e-78)
     (* (/ 0.3333333333333333 a) (- (sqrt (* a (* c -3.0))) b))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.66e-195) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 4.8e-78) {
		tmp = (0.3333333333333333 / a) * (sqrt((a * (c * -3.0))) - b);
	} else {
		tmp = -0.5 * (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 <= (-1.66d-195)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 4.8d-78) then
        tmp = (0.3333333333333333d0 / a) * (sqrt((a * (c * (-3.0d0)))) - b)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.66e-195) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 4.8e-78) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt((a * (c * -3.0))) - b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.66e-195:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 4.8e-78:
		tmp = (0.3333333333333333 / a) * (math.sqrt((a * (c * -3.0))) - b)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.66e-195)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 4.8e-78)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(a * Float64(c * -3.0))) - b));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.66e-195)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 4.8e-78)
		tmp = (0.3333333333333333 / a) * (sqrt((a * (c * -3.0))) - b);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.66e-195], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.8e-78], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.66e-195

   1. Initial program 71.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 84.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -1.66e-195 < b < 4.79999999999999999e-78

   1. Initial program 71.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around 0 69.1%

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

      1. expm1-log1p-u 43.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 19.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\right)} - 1} \]

      3. neg-mul-1 19.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot b} + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\right)} - 1 \]

      4. fma-def 19.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{-3 \cdot \left(c \cdot a\right)}\right)}}{3 \cdot a}\right)} - 1 \]

      5. *-commutative 19.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{-3 \cdot \color{blue}{\left(a \cdot c\right)}}\right)}{3 \cdot a}\right)} - 1 \]

      6. *-commutative 19.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}\right)}{3 \cdot a}\right)} - 1 \]

      7. *-commutative 19.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3}\right)}{3 \cdot a}\right)} - 1 \]

      8. associate-*r* 20.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}\right)}{3 \cdot a}\right)} - 1 \]

   4. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{3 \cdot a}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 43.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{3 \cdot a}\right)\right)} \]

      2. expm1-log1p 69.3%

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

      3. *-lft-identity 69.3%

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

      4. associate-*l/ 69.2%

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

      5. associate-/r* 69.1%

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

      6. metadata-eval 69.1%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{c \cdot \left(a \cdot -3\right)}\right) \]

      7. fma-udef 69.1%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-1 \cdot b + \sqrt{c \cdot \left(a \cdot -3\right)}\right)} \]

      8. neg-mul-1 69.1%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\left(-b\right)} + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \]

      9. +-commutative 69.1%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(\sqrt{c \cdot \left(a \cdot -3\right)} + \left(-b\right)\right)} \]

      10. unsub-neg 69.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(\sqrt{c \cdot \left(a \cdot -3\right)} - b\right)} \]

      11. associate-*r* 69.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\sqrt{\color{blue}{\left(c \cdot a\right) \cdot -3}} - b\right) \]

      12. *-commutative 69.1%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -3} - b\right) \]

      13. associate-*l* 69.2%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}} - b\right) \]

   6. Simplified 69.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)} \]

   if 4.79999999999999999e-78 < b

   1. Initial program 17.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 87.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.66 \cdot 10^{-195}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.66 \cdot 10^{-195}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.66e-195)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 1.25e-77)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.66e-195) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.25e-77) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (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 <= (-1.66d-195)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 1.25d-77) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.66e-195) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 1.25e-77) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.66e-195:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 1.25e-77:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.66e-195)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 1.25e-77)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.66e-195)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 1.25e-77)
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.66e-195], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-77], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.66e-195

   1. Initial program 71.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 84.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -1.66e-195 < b < 1.24999999999999991e-77

   1. Initial program 71.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around 0 69.1%

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

      1. *-commutative 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

      4. associate-*l* 69.3%

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

   4. Simplified 69.3%

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

   if 1.24999999999999991e-77 < b

   1. Initial program 17.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 87.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.66 \cdot 10^{-195}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 68.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = -0.5 * (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 <= (-4d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 70.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 30.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 71.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8: 68.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{1}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* b (/ 1.0 (* a -1.5))) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (1.0 / (a * -1.5));
	} else {
		tmp = -0.5 * (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 <= (-4d-310)) then
        tmp = b * (1.0d0 / (a * (-1.5d0)))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (1.0 / (a * -1.5));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = b * (1.0 / (a * -1.5))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(b * Float64(1.0 / Float64(a * -1.5)));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = b * (1.0 / (a * -1.5));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(b * N[(1.0 / N[(a * -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. /-rgt-identity 72.0%

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

      2. metadata-eval 72.0%

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

      3. associate-/r/ 72.0%

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

      4. metadata-eval 72.0%

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

      5. metadata-eval 72.0%

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

      6. times-frac 72.0%

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

      7. *-commutative 72.0%

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

      8. times-frac 71.9%

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

      9. *-commutative 71.9%

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

      10. associate-/r* 71.8%

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

      11. associate-*l/ 71.8%

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

   3. Simplified 71.8%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]
   4. Step-by-step derivation

      1. clear-num 71.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}}} \]

      2. inv-pow 71.8%

         \[\leadsto \color{blue}{{\left(\frac{a}{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}\right)}^{-1}} \]

      3. *-commutative 71.8%

         \[\leadsto {\left(\frac{a}{\color{blue}{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot -0.3333333333333333}}\right)}^{-1} \]

      4. fma-udef 71.8%

         \[\leadsto {\left(\frac{a}{\left(b - \sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -3}}\right) \cdot -0.3333333333333333}\right)}^{-1} \]

      5. associate-*r* 71.8%

         \[\leadsto {\left(\frac{a}{\left(b - \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -3\right)}}\right) \cdot -0.3333333333333333}\right)}^{-1} \]

      6. add-sqr-sqrt 56.6%

         \[\leadsto {\left(\frac{a}{\left(b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}}}\right) \cdot -0.3333333333333333}\right)}^{-1} \]

      7. hypot-def 66.6%

         \[\leadsto {\left(\frac{a}{\left(b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}\right) \cdot -0.3333333333333333}\right)}^{-1} \]

   5. Applied egg-rr 66.6%

      \[\leadsto \color{blue}{{\left(\frac{a}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot -0.3333333333333333}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 66.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot -0.3333333333333333}}} \]

      2. *-commutative 66.6%

         \[\leadsto \frac{1}{\frac{a}{\color{blue}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}}} \]

      3. associate-*r* 66.6%

         \[\leadsto \frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}\right)\right)}} \]

      4. *-commutative 66.6%

         \[\leadsto \frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3}\right)\right)}} \]

      5. associate-*l* 66.6%

         \[\leadsto \frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}\right)\right)}} \]

   7. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right)}}} \]

   8. Taylor expanded in b around -inf 70.3%

      \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]
   9. Step-by-step derivation

      1. *-commutative 70.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} \cdot -1.5}} \]

      2. associate-*l/ 70.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot -1.5}{b}}} \]

   10. Simplified 70.3%

       \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot -1.5}{b}}} \]
   11. Step-by-step derivation

       1. associate-/r/ 70.4%

          \[\leadsto \color{blue}{\frac{1}{a \cdot -1.5} \cdot b} \]

   12. Applied egg-rr 70.4%

       \[\leadsto \color{blue}{\frac{1}{a \cdot -1.5} \cdot b} \]

   if -3.999999999999988e-310 < b

   1. Initial program 30.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 71.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{1}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9: 68.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* (/ b 3.0) (/ -2.0 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else {
		tmp = -0.5 * (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 <= (-4d-310)) then
        tmp = (b / 3.0d0) * ((-2.0d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (b / 3.0) * (-2.0 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (b / 3.0) * (-2.0 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(b / 3.0) * Float64(-2.0 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = (b / 3.0) * (-2.0 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(b / 3.0), $MachinePrecision] * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 70.4%

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

      1. *-commutative 70.4%

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

   4. Simplified 70.4%

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

      1. times-frac 70.4%

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

   6. Applied egg-rr 70.4%

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

   if -3.999999999999988e-310 < b

   1. Initial program 30.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 71.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{3} \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 68.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (/ (* b -2.0) (* 3.0 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = -0.5 * (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 <= (-4d-310)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (b * -2.0) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = (b * -2.0) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 70.4%

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

      1. *-commutative 70.4%

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

   4. Simplified 70.4%

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

   if -3.999999999999988e-310 < b

   1. Initial program 30.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 71.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11: 68.3% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (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 <= (-4d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 70.4%

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

      1. *-commutative 70.4%

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

   4. Simplified 70.4%

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

      1. expm1-log1p-u 48.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b \cdot -2}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 33.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b \cdot -2}{3 \cdot a}\right)} - 1} \]

      3. times-frac 33.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{b}{3} \cdot \frac{-2}{a}}\right)} - 1 \]

   6. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b}{3} \cdot \frac{-2}{a}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 48.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{3} \cdot \frac{-2}{a}\right)\right)} \]

      2. expm1-log1p 70.4%

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

      3. associate-*r/ 70.4%

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

      4. associate-*l/ 70.4%

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

      5. associate-/r* 70.4%

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

      6. *-rgt-identity 70.4%

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

      7. associate-*r/ 70.4%

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

      8. associate-*l* 70.4%

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

      9. associate-/r* 70.3%

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

      10. metadata-eval 70.3%

          \[\leadsto b \cdot \left(-2 \cdot \frac{\color{blue}{0.3333333333333333}}{a}\right) \]

      11. associate-*r/ 70.3%

          \[\leadsto b \cdot \color{blue}{\frac{-2 \cdot 0.3333333333333333}{a}} \]

      12. metadata-eval 70.3%

          \[\leadsto b \cdot \frac{\color{blue}{-0.6666666666666666}}{a} \]

   8. Simplified 70.3%

      \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 30.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 71.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12: 36.1% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (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 = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in b around inf 31.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]

3. Final simplification 31.8%

   \[\leadsto -0.5 \cdot \frac{c}{b} \]

Reproduce

herbie shell --seed 2023213 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))