Cubic critical

Percentage Accurate: 52.9% → 85.1%

Time: 13.4s

Alternatives: 11

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: 52.9% 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: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e+143)
   (/ (/ b -1.5) a)
   (if (<= b 1.42e-98)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+143) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.42e-98) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / 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 <= (-8.5d+143)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 1.42d-98) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e+143) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.42e-98) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e+143:
		tmp = (b / -1.5) / a
	elif b <= 1.42e-98:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e+143)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 1.42e-98)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e+143)
		tmp = (b / -1.5) / a;
	elseif (b <= 1.42e-98)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e+143], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.42e-98], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999998e143

   1. Initial program 45.4%

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

   2. Taylor expanded in b around -inf 99.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 99.6%

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

   4. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   5. Step-by-step derivation

      1. *-commutative 99.6%

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

      2. clear-num 99.6%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 99.7%

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

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

         \[\leadsto \frac{-0.6666666666666666 \cdot b}{\color{blue}{1 \cdot a}} \]

      3. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{1} \cdot \frac{b}{a}} \]

      4. metadata-eval 99.6%

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

      5. metadata-eval 99.6%

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

      6. times-frac 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-/r* 99.9%

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

      9. associate-/l* 99.9%

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

      10. metadata-eval 99.9%

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

   10. Applied egg-rr 99.9%

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

   if -8.4999999999999998e143 < b < 1.41999999999999999e-98

   1. Initial program 86.8%

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

   if 1.41999999999999999e-98 < b

   1. Initial program 19.0%

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

   2. Taylor expanded in b around inf 93.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 93.5%

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

   4. Simplified 93.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \left(\frac{c}{b} \cdot -3\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-70)
   (fma b (/ -0.6666666666666666 a) (* (* (/ c b) -3.0) -0.16666666666666666))
   (if (<= b 6.5e-99)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-70) {
		tmp = fma(b, (-0.6666666666666666 / a), (((c / b) * -3.0) * -0.16666666666666666));
	} else if (b <= 6.5e-99) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-70)
		tmp = fma(b, Float64(-0.6666666666666666 / a), Float64(Float64(Float64(c / b) * -3.0) * -0.16666666666666666));
	elseif (b <= 6.5e-99)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-70], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision] + N[(N[(N[(c / b), $MachinePrecision] * -3.0), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-99], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.99999999999999996e-71

   1. Initial program 73.7%

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

   3. Applied egg-rr 61.7%

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

      1. div-inv 61.6%

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

      2. unpow-prod-down 61.6%

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

      3. inv-pow 61.6%

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

      4. clear-num 61.6%

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

      5. inv-pow 61.6%

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

      6. clear-num 61.7%

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

      7. times-frac 61.7%

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

      8. *-un-lft-identity 61.7%

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

      9. associate-/r* 61.6%

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

      10. frac-2neg 61.6%

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

      11. metadata-eval 61.6%

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

   5. Applied egg-rr 61.6%

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

      1. distribute-neg-frac 61.6%

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

      2. sub-neg 61.6%

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

      3. +-commutative 61.6%

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

      4. distribute-neg-in 61.6%

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

      5. remove-double-neg 61.6%

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

      6. sub-neg 61.6%

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

   7. Simplified 61.6%

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

   8. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   9. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]

      2. associate-*l/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]

      3. *-commutative 0.0%

         \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]

      4. fma-def 0.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}\right)} \]

      5. *-commutative 0.0%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \color{blue}{\frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \cdot -0.16666666666666666}\right) \]

      6. associate-/l* 0.0%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \color{blue}{\frac{c}{\frac{b}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot -0.16666666666666666\right) \]

      7. unpow2 0.0%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{c}{\frac{b}{\color{blue}{\sqrt{-3} \cdot \sqrt{-3}}}} \cdot -0.16666666666666666\right) \]

      8. rem-square-sqrt 92.2%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{c}{\frac{b}{\color{blue}{-3}}} \cdot -0.16666666666666666\right) \]

      9. associate-/r/ 92.2%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \color{blue}{\left(\frac{c}{b} \cdot -3\right)} \cdot -0.16666666666666666\right) \]

   10. Simplified 92.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \left(\frac{c}{b} \cdot -3\right) \cdot -0.16666666666666666\right)} \]

   if -9.99999999999999996e-71 < b < 6.50000000000000033e-99

   1. Initial program 76.5%

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

   2. Taylor expanded in b around 0 70.6%

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

   if 6.50000000000000033e-99 < b

   1. Initial program 19.0%

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

   2. Taylor expanded in b around inf 93.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 93.5%

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

   4. Simplified 93.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \left(\frac{c}{b} \cdot -3\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \left(\frac{c}{b} \cdot -3\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.25e-70)
   (fma b (/ -0.6666666666666666 a) (* (* (/ c b) -3.0) -0.16666666666666666))
   (if (<= b 1.05e-98)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.25e-70) {
		tmp = fma(b, (-0.6666666666666666 / a), (((c / b) * -3.0) * -0.16666666666666666));
	} else if (b <= 1.05e-98) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.25e-70)
		tmp = fma(b, Float64(-0.6666666666666666 / a), Float64(Float64(Float64(c / b) * -3.0) * -0.16666666666666666));
	elseif (b <= 1.05e-98)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.25e-70], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision] + N[(N[(N[(c / b), $MachinePrecision] * -3.0), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e-98], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.25e-70

   1. Initial program 73.7%

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

   3. Applied egg-rr 61.7%

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

      1. div-inv 61.6%

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

      2. unpow-prod-down 61.6%

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

      3. inv-pow 61.6%

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

      4. clear-num 61.6%

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

      5. inv-pow 61.6%

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

      6. clear-num 61.7%

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

      7. times-frac 61.7%

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

      8. *-un-lft-identity 61.7%

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

      9. associate-/r* 61.6%

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

      10. frac-2neg 61.6%

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

      11. metadata-eval 61.6%

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

   5. Applied egg-rr 61.6%

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

      1. distribute-neg-frac 61.6%

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

      2. sub-neg 61.6%

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

      3. +-commutative 61.6%

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

      4. distribute-neg-in 61.6%

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

      5. remove-double-neg 61.6%

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

      6. sub-neg 61.6%

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

   7. Simplified 61.6%

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

   8. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   9. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]

      2. associate-*l/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]

      3. *-commutative 0.0%

         \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]

      4. fma-def 0.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}\right)} \]

      5. *-commutative 0.0%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \color{blue}{\frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \cdot -0.16666666666666666}\right) \]

      6. associate-/l* 0.0%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \color{blue}{\frac{c}{\frac{b}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot -0.16666666666666666\right) \]

      7. unpow2 0.0%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{c}{\frac{b}{\color{blue}{\sqrt{-3} \cdot \sqrt{-3}}}} \cdot -0.16666666666666666\right) \]

      8. rem-square-sqrt 92.2%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{c}{\frac{b}{\color{blue}{-3}}} \cdot -0.16666666666666666\right) \]

      9. associate-/r/ 92.2%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \color{blue}{\left(\frac{c}{b} \cdot -3\right)} \cdot -0.16666666666666666\right) \]

   10. Simplified 92.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \left(\frac{c}{b} \cdot -3\right) \cdot -0.16666666666666666\right)} \]

   if -1.25e-70 < b < 1.04999999999999996e-98

   1. Initial program 76.5%

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

   2. Taylor expanded in b around 0 70.6%

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

      1. associate-*r* 70.7%

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

      2. *-commutative 70.7%

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

   4. Simplified 70.7%

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

   if 1.04999999999999996e-98 < b

   1. Initial program 19.0%

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

   2. Taylor expanded in b around inf 93.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 93.5%

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

   4. Simplified 93.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \left(\frac{c}{b} \cdot -3\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \left(\frac{c}{b} \cdot -3\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-312)
   (fma b (/ -0.6666666666666666 a) (* (* (/ c b) -3.0) -0.16666666666666666))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = fma(b, (-0.6666666666666666 / a), (((c / b) * -3.0) * -0.16666666666666666));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-312)
		tmp = fma(b, Float64(-0.6666666666666666 / a), Float64(Float64(Float64(c / b) * -3.0) * -0.16666666666666666));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-312], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision] + N[(N[(N[(c / b), $MachinePrecision] * -3.0), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.0000000000022e-312

   1. Initial program 75.7%

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

   3. Applied egg-rr 66.1%

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

      1. div-inv 66.0%

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

      2. unpow-prod-down 66.0%

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

      3. inv-pow 66.0%

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

      4. clear-num 66.0%

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

      5. inv-pow 66.0%

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

      6. clear-num 66.0%

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

      7. times-frac 66.1%

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

      8. *-un-lft-identity 66.1%

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

      9. associate-/r* 66.1%

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

      10. frac-2neg 66.1%

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

      11. metadata-eval 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. distribute-neg-frac 66.1%

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

      2. sub-neg 66.1%

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

      3. +-commutative 66.1%

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

      4. distribute-neg-in 66.1%

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

      5. remove-double-neg 66.1%

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

      6. sub-neg 66.1%

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

   7. Simplified 66.1%

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

   8. Taylor expanded in b around -inf 0.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   9. Step-by-step derivation

      1. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]

      2. associate-*l/ 0.0%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]

      3. *-commutative 0.0%

         \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} + -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]

      4. fma-def 0.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, -0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}\right)} \]

      5. *-commutative 0.0%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \color{blue}{\frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \cdot -0.16666666666666666}\right) \]

      6. associate-/l* 0.0%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \color{blue}{\frac{c}{\frac{b}{{\left(\sqrt{-3}\right)}^{2}}}} \cdot -0.16666666666666666\right) \]

      7. unpow2 0.0%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{c}{\frac{b}{\color{blue}{\sqrt{-3} \cdot \sqrt{-3}}}} \cdot -0.16666666666666666\right) \]

      8. rem-square-sqrt 71.8%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{c}{\frac{b}{\color{blue}{-3}}} \cdot -0.16666666666666666\right) \]

      9. associate-/r/ 71.8%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \color{blue}{\left(\frac{c}{b} \cdot -3\right)} \cdot -0.16666666666666666\right) \]

   10. Simplified 71.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \left(\frac{c}{b} \cdot -3\right) \cdot -0.16666666666666666\right)} \]

   if -5.0000000000022e-312 < b

   1. Initial program 28.4%

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

   2. Taylor expanded in b around inf 77.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 77.3%

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

   4. Simplified 77.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \left(\frac{c}{b} \cdot -3\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 67.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-312)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c * -0.5) / 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 <= (-5d-312)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-312:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-312)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-312)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-312], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.0000000000022e-312

   1. Initial program 75.7%

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

   2. Taylor expanded in b around -inf 71.7%

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

   if -5.0000000000022e-312 < b

   1. Initial program 28.4%

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

   2. Taylor expanded in b around inf 77.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 77.3%

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

   4. Simplified 77.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 43.2% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+75}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.5e+75) (* b (/ -0.6666666666666666 a)) (* (/ c b) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e+75) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	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 <= 2.5d+75) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e+75) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.5e+75:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.5e+75)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.5e+75)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.5e+75], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5000000000000001e75

   1. Initial program 65.0%

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

   2. Taylor expanded in b around -inf 51.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 51.9%

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

   4. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   5. Step-by-step derivation

      1. *-commutative 51.9%

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

      2. clear-num 51.9%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 51.9%

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

   6. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 51.9%

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

   8. Simplified 51.9%

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

   if 2.5000000000000001e75 < b

   1. Initial program 19.7%

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

   2. Taylor expanded in b around -inf 2.3%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

   3. Taylor expanded in b around 0 42.6%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+75}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \]

Alternative 7: 43.2% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.5e+75) (/ -0.6666666666666666 (/ a b)) (* (/ c b) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e+75) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c / b) * 0.5;
	}
	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 <= 2.5d+75) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (c / b) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e+75) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.5e+75:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = (c / b) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.5e+75)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(Float64(c / b) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.5e+75)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = (c / b) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.5e+75], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5000000000000001e75

   1. Initial program 65.0%

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

   2. Taylor expanded in b around -inf 51.9%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 51.9%

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

   4. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   5. Step-by-step derivation

      1. *-commutative 51.9%

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

      2. clear-num 51.9%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 51.9%

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

   6. Applied egg-rr 51.9%

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

   if 2.5000000000000001e75 < b

   1. Initial program 19.7%

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

   2. Taylor expanded in b around -inf 2.3%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

   3. Taylor expanded in b around 0 42.6%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \]

Alternative 8: 67.0% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-312) (/ -0.6666666666666666 (/ a b)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-312)) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-312:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-312)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-312)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-312], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.0000000000022e-312

   1. Initial program 75.7%

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

   2. Taylor expanded in b around -inf 71.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 71.2%

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

   4. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   5. Step-by-step derivation

      1. *-commutative 71.2%

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

      2. clear-num 71.2%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 71.3%

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

   6. Applied egg-rr 71.3%

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

   if -5.0000000000022e-312 < b

   1. Initial program 28.4%

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

   2. Taylor expanded in b around inf 77.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 77.3%

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

      2. associate-/l* 76.5%

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

   4. Simplified 76.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]

Alternative 9: 67.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-312) (/ b (* -1.5 a)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-312)) then
        tmp = b / ((-1.5d0) * a)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-312:
		tmp = b / (-1.5 * a)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-312)
		tmp = Float64(b / Float64(-1.5 * a));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-312)
		tmp = b / (-1.5 * a);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-312], N[(b / N[(-1.5 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.0000000000022e-312

   1. Initial program 75.7%

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

   2. Taylor expanded in b around -inf 71.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 71.2%

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

   4. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   5. Step-by-step derivation

      1. *-commutative 71.2%

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

      2. clear-num 71.2%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 71.3%

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

   6. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 71.2%

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

   8. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]
   9. Step-by-step derivation

      1. *-commutative 71.2%

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

      2. clear-num 71.2%

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

      3. un-div-inv 71.3%

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

      4. div-inv 71.4%

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

      5. metadata-eval 71.4%

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

   10. Applied egg-rr 71.4%

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

   if -5.0000000000022e-312 < b

   1. Initial program 28.4%

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

   2. Taylor expanded in b around inf 77.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 77.3%

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

      2. associate-/l* 76.5%

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

   4. Simplified 76.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]

Alternative 10: 67.4% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-312) (/ b (* -1.5 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = (c * -0.5) / 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 <= (-5d-312)) then
        tmp = b / ((-1.5d0) * a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-312) {
		tmp = b / (-1.5 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-312:
		tmp = b / (-1.5 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-312)
		tmp = Float64(b / Float64(-1.5 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-312)
		tmp = b / (-1.5 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-312], N[(b / N[(-1.5 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.0000000000022e-312

   1. Initial program 75.7%

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

   2. Taylor expanded in b around -inf 71.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. *-commutative 71.2%

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

   4. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   5. Step-by-step derivation

      1. *-commutative 71.2%

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

      2. clear-num 71.2%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 71.3%

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

   6. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 71.2%

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

   8. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]
   9. Step-by-step derivation

      1. *-commutative 71.2%

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

      2. clear-num 71.2%

         \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

      3. un-div-inv 71.3%

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

      4. div-inv 71.4%

         \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

      5. metadata-eval 71.4%

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

   10. Applied egg-rr 71.4%

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

   if -5.0000000000022e-312 < b

   1. Initial program 28.4%

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

   2. Taylor expanded in b around inf 77.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. associate-*r/ 77.3%

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

   4. Simplified 77.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{b}{-1.5 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 11: 11.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

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) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in b around -inf 39.0%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

3. Taylor expanded in b around 0 12.8%

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

4. Final simplification 12.8%

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

Reproduce

herbie shell --seed 2023306 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))