Cubic critical

Percentage Accurate: 51.7% → 85.9%

Time: 4.5s

Alternatives: 15

Speedup: 2.2×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    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: 51.7% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    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: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e+125)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 4.4e-78)
     (/ (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) 3.0) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e+125) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = ((sqrt(fma((c * -3.0), a, (b * b))) - b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e+125)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 4.4e-78)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / 3.0) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e+125], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-78], N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.60000000000000003e125

   1. Initial program 41.6%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -2.60000000000000003e125 < b < 4.3999999999999998e-78

   1. Initial program 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 79.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lift-neg.f64 79.9

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

   6. Applied rewrites 79.9%

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

   if 4.3999999999999998e-78 < b

   1. Initial program 17.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+125}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+154)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 4.4e-78)
     (/ (- (sqrt (fma (* -3.0 a) c (* b b))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+154) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+154)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 4.4e-78)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+154], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-78], N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000004e154

   1. Initial program 34.5%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if -1.00000000000000004e154 < b < 4.3999999999999998e-78

   1. Initial program 80.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 80.8%

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

   if 4.3999999999999998e-78 < b

   1. Initial program 17.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(a \cdot -3\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-95)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 4.4e-78)
     (/ (+ (- b) (sqrt (* (* a -3.0) c))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-95) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = (-b + sqrt(((a * -3.0) * c))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.2d-95)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 4.4d-78) then
        tmp = (-b + sqrt(((a * (-3.0d0)) * c))) / (3.0d0 * 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 <= -3.2e-95) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = (-b + Math.sqrt(((a * -3.0) * c))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.2e-95:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 4.4e-78:
		tmp = (-b + math.sqrt(((a * -3.0) * c))) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-95)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 4.4e-78)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(a * -3.0) * c))) / Float64(3.0 * 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 <= -3.2e-95)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 4.4e-78)
		tmp = (-b + sqrt(((a * -3.0) * c))) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-95], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-78], N[(N[((-b) + N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1999999999999997e-95

   1. Initial program 62.9%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -3.1999999999999997e-95 < b < 4.3999999999999998e-78

   1. Initial program 75.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 72.4

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

   7. Applied rewrites 72.4%

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

   if 4.3999999999999998e-78 < b

   1. Initial program 17.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

Alternative 4: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-95)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 4.4e-78)
     (/ (+ (- b) (sqrt (* -3.0 (* c a)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-95) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = (-b + sqrt((-3.0 * (c * a)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.2d-95)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 4.4d-78) then
        tmp = (-b + sqrt(((-3.0d0) * (c * a)))) / (3.0d0 * 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 <= -3.2e-95) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = (-b + Math.sqrt((-3.0 * (c * a)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.2e-95:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 4.4e-78:
		tmp = (-b + math.sqrt((-3.0 * (c * a)))) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-95)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 4.4e-78)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(-3.0 * Float64(c * a)))) / Float64(3.0 * 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 <= -3.2e-95)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 4.4e-78)
		tmp = (-b + sqrt((-3.0 * (c * a)))) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-95], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-78], N[(N[((-b) + N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1999999999999997e-95

   1. Initial program 62.9%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -3.1999999999999997e-95 < b < 4.3999999999999998e-78

   1. Initial program 75.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 4.3999999999999998e-78 < b

   1. Initial program 17.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-120}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -3\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-120)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 4.4e-78) (/ (sqrt (* (* a -3.0) c)) (* 3.0 a)) (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-120) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = sqrt(((a * -3.0) * c)) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-120)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 4.4d-78) then
        tmp = sqrt(((a * (-3.0d0)) * c)) / (3.0d0 * 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 <= -1e-120) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = Math.sqrt(((a * -3.0) * c)) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-120:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 4.4e-78:
		tmp = math.sqrt(((a * -3.0) * c)) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-120)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 4.4e-78)
		tmp = Float64(sqrt(Float64(Float64(a * -3.0) * c)) / Float64(3.0 * 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 <= -1e-120)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 4.4e-78)
		tmp = sqrt(((a * -3.0) * c)) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-120], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-78], N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.99999999999999979e-121

   1. Initial program 63.8%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if -9.99999999999999979e-121 < b < 4.3999999999999998e-78

   1. Initial program 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 74.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lift-neg.f64 74.2

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

   6. Applied rewrites 74.2%

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

   7. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 71.7

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

   9. Applied rewrites 71.7%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lower-/.f64 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. *-commutative N/A

          \[\leadsto \frac{\sqrt{\left(a \cdot -3\right) \cdot c}}{\mathsf{Rewrite<=}\left(lift-*.f64, \left(3 \cdot a\right)\right)} \]

   11. Applied rewrites 71.7%

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

   if 4.3999999999999998e-78 < b

   1. Initial program 17.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-120}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-120)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 4.4e-78) (/ (sqrt (* -3.0 (* c a))) (* 3.0 a)) (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-120) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = sqrt((-3.0 * (c * a))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-120)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 4.4d-78) then
        tmp = sqrt(((-3.0d0) * (c * a))) / (3.0d0 * 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 <= -1e-120) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 4.4e-78) {
		tmp = Math.sqrt((-3.0 * (c * a))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-120:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 4.4e-78:
		tmp = math.sqrt((-3.0 * (c * a))) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-120)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 4.4e-78)
		tmp = Float64(sqrt(Float64(-3.0 * Float64(c * a))) / Float64(3.0 * 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 <= -1e-120)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 4.4e-78)
		tmp = sqrt((-3.0 * (c * a))) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-120], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-78], N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.99999999999999979e-121

   1. Initial program 63.8%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if -9.99999999999999979e-121 < b < 4.3999999999999998e-78

   1. Initial program 74.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\sqrt{a \cdot c} \cdot \sqrt{-3}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 71.6

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

   5. Applied rewrites 71.6%

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

   if 4.3999999999999998e-78 < b

   1. Initial program 17.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 85.2

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

   5. Applied rewrites 85.2%

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

Alternative 7: 72.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-133}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-133)
   (/ (* -2.0 b) (* 3.0 a))
   (if (<= b 3.6e-119)
     (* (sqrt (* (/ c a) -3.0)) 0.3333333333333333)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-133) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 3.6e-119) {
		tmp = sqrt(((c / a) * -3.0)) * 0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8d-133)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * a)
    else if (b <= 3.6d-119) then
        tmp = sqrt(((c / a) * (-3.0d0))) * 0.3333333333333333d0
    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 <= -8e-133) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else if (b <= 3.6e-119) {
		tmp = Math.sqrt(((c / a) * -3.0)) * 0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8e-133:
		tmp = (-2.0 * b) / (3.0 * a)
	elif b <= 3.6e-119:
		tmp = math.sqrt(((c / a) * -3.0)) * 0.3333333333333333
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e-133)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * a));
	elseif (b <= 3.6e-119)
		tmp = Float64(sqrt(Float64(Float64(c / a) * -3.0)) * 0.3333333333333333);
	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 <= -8e-133)
		tmp = (-2.0 * b) / (3.0 * a);
	elseif (b <= 3.6e-119)
		tmp = sqrt(((c / a) * -3.0)) * 0.3333333333333333;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e-133], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e-119], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.0000000000000005e-133

   1. Initial program 65.0%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -8.0000000000000005e-133 < b < 3.6e-119

   1. Initial program 72.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{1}{3} \]

      10. lower-/.f64 44.8

          \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333 \]

   5. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a} \cdot -3} \cdot 0.3333333333333333} \]

   if 3.6e-119 < b

   1. Initial program 20.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.6

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

   5. Applied rewrites 81.6%

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

Alternative 8: 70.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{3}}{a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\frac{c}{a} \cdot -3} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-266)
   (/ (/ (* -2.0 b) 3.0) a)
   (if (<= b 9.8e-109)
     (* (sqrt (* (/ c a) -3.0)) -0.3333333333333333)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-266) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 9.8e-109) {
		tmp = sqrt(((c / a) * -3.0)) * -0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.2d-266)) then
        tmp = (((-2.0d0) * b) / 3.0d0) / a
    else if (b <= 9.8d-109) then
        tmp = sqrt(((c / a) * (-3.0d0))) * (-0.3333333333333333d0)
    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 <= -4.2e-266) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 9.8e-109) {
		tmp = Math.sqrt(((c / a) * -3.0)) * -0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-266:
		tmp = ((-2.0 * b) / 3.0) / a
	elif b <= 9.8e-109:
		tmp = math.sqrt(((c / a) * -3.0)) * -0.3333333333333333
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-266)
		tmp = Float64(Float64(Float64(-2.0 * b) / 3.0) / a);
	elseif (b <= 9.8e-109)
		tmp = Float64(sqrt(Float64(Float64(c / a) * -3.0)) * -0.3333333333333333);
	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 <= -4.2e-266)
		tmp = ((-2.0 * b) / 3.0) / a;
	elseif (b <= 9.8e-109)
		tmp = sqrt(((c / a) * -3.0)) * -0.3333333333333333;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-266], N[(N[(N[(-2.0 * b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 9.8e-109], N[(N[Sqrt[N[(N[(c / a), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.19999999999999994e-266

   1. Initial program 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 72.7

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

   7. Applied rewrites 72.7%

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

   if -4.19999999999999994e-266 < b < 9.79999999999999997e-109

   1. Initial program 76.8%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{-1}{3} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{-1}{3} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{-1}{3} \]

      8. lower-/.f64 28.7

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot -0.3333333333333333 \]

   5. Applied rewrites 28.7%

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a} \cdot -3} \cdot -0.3333333333333333} \]

   if 9.79999999999999997e-109 < b

   1. Initial program 18.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

Alternative 9: 70.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{3}}{a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{c \cdot \frac{-3}{a}} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-266)
   (/ (/ (* -2.0 b) 3.0) a)
   (if (<= b 9.8e-109)
     (* (sqrt (* c (/ -3.0 a))) -0.3333333333333333)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-266) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 9.8e-109) {
		tmp = sqrt((c * (-3.0 / a))) * -0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.2d-266)) then
        tmp = (((-2.0d0) * b) / 3.0d0) / a
    else if (b <= 9.8d-109) then
        tmp = sqrt((c * ((-3.0d0) / a))) * (-0.3333333333333333d0)
    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 <= -4.2e-266) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else if (b <= 9.8e-109) {
		tmp = Math.sqrt((c * (-3.0 / a))) * -0.3333333333333333;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-266:
		tmp = ((-2.0 * b) / 3.0) / a
	elif b <= 9.8e-109:
		tmp = math.sqrt((c * (-3.0 / a))) * -0.3333333333333333
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.2e-266)
		tmp = Float64(Float64(Float64(-2.0 * b) / 3.0) / a);
	elseif (b <= 9.8e-109)
		tmp = Float64(sqrt(Float64(c * Float64(-3.0 / a))) * -0.3333333333333333);
	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 <= -4.2e-266)
		tmp = ((-2.0 * b) / 3.0) / a;
	elseif (b <= 9.8e-109)
		tmp = sqrt((c * (-3.0 / a))) * -0.3333333333333333;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.2e-266], N[(N[(N[(-2.0 * b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 9.8e-109], N[(N[Sqrt[N[(c * N[(-3.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.19999999999999994e-266

   1. Initial program 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 72.7

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

   7. Applied rewrites 72.7%

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

   if -4.19999999999999994e-266 < b < 9.79999999999999997e-109

   1. Initial program 76.8%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{-1}{3} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{-1}{3} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{-1}{3} \]

      8. lower-/.f64 28.7

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot -0.3333333333333333 \]

   5. Applied rewrites 28.7%

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a} \cdot -3} \cdot -0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{-1}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -3} \cdot \frac{-1}{3} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{-3 \cdot \frac{c}{a}} \cdot \frac{-1}{3} \]

      4. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{-3 \cdot c}{a}} \cdot \frac{-1}{3} \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{-3 \cdot c}{a}} \cdot \frac{-1}{3} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot \frac{-1}{3} \]

      7. lower-*.f64 28.6

         \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot -0.3333333333333333 \]

   7. Applied rewrites 28.6%

      \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot -0.3333333333333333 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot \frac{-1}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{c \cdot -3}{a}} \cdot \frac{-1}{3} \]

      3. associate-/l* N/A

         \[\leadsto \sqrt{c \cdot \frac{-3}{a}} \cdot \frac{-1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{c \cdot \frac{-3}{a}} \cdot \frac{-1}{3} \]

      5. lower-/.f64 28.6

         \[\leadsto \sqrt{c \cdot \frac{-3}{a}} \cdot -0.3333333333333333 \]

   9. Applied rewrites 28.6%

      \[\leadsto \sqrt{c \cdot \frac{-3}{a}} \cdot -0.3333333333333333 \]

   if 9.79999999999999997e-109 < b

   1. Initial program 18.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.7

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

   5. Applied rewrites 83.7%

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

Alternative 10: 68.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{-2 \cdot b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-311) (/ (/ (* -2.0 b) 3.0) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-311) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-311)) then
        tmp = (((-2.0d0) * b) / 3.0d0) / 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 <= -4e-311) {
		tmp = ((-2.0 * b) / 3.0) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-311:
		tmp = ((-2.0 * b) / 3.0) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-311)
		tmp = Float64(Float64(Float64(-2.0 * b) / 3.0) / 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 <= -4e-311)
		tmp = ((-2.0 * b) / 3.0) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-311], N[(N[(N[(-2.0 * b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999979e-311

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 68.1

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

   7. Applied rewrites 68.1%

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

   if -3.99999999999979e-311 < b

   1. Initial program 29.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 69.2

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

   5. Applied rewrites 69.2%

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

Alternative 11: 68.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{-2 \cdot b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-311) (/ (* -2.0 b) (* 3.0 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-311) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-311)) then
        tmp = ((-2.0d0) * b) / (3.0d0 * 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 <= -4e-311) {
		tmp = (-2.0 * b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-311:
		tmp = (-2.0 * b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-311)
		tmp = Float64(Float64(-2.0 * b) / Float64(3.0 * 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 <= -4e-311)
		tmp = (-2.0 * b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-311], N[(N[(-2.0 * b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999979e-311

   1. Initial program 66.9%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if -3.99999999999979e-311 < b

   1. Initial program 29.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 69.2

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

   5. Applied rewrites 69.2%

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

Alternative 12: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{-0.6666666666666666 \cdot b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-311) (/ (* -0.6666666666666666 b) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-311) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-311)) then
        tmp = ((-0.6666666666666666d0) * b) / 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 <= -4e-311) {
		tmp = (-0.6666666666666666 * b) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-311:
		tmp = (-0.6666666666666666 * b) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-311)
		tmp = Float64(Float64(-0.6666666666666666 * b) / 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 <= -4e-311)
		tmp = (-0.6666666666666666 * b) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-311], N[(N[(-0.6666666666666666 * b), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999979e-311

   1. Initial program 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 67.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if -3.99999999999979e-311 < b

   1. Initial program 29.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 69.2

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

   5. Applied rewrites 69.2%

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

Alternative 13: 68.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-311}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-311) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-311) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-311)) then
        tmp = (-0.6666666666666666d0) * (b / 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 <= -4e-311) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-311:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-311)
		tmp = Float64(-0.6666666666666666 * Float64(b / 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 <= -4e-311)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-311], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999979e-311

   1. Initial program 66.9%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if -3.99999999999979e-311 < b

   1. Initial program 29.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 69.2

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

   5. Applied rewrites 69.2%

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

Alternative 14: 35.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.6666666666666666 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.6666666666666666 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.6666666666666666d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Python

def code(a, b, c):
	return -0.6666666666666666 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.6666666666666666 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.6666666666666666 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.9%

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

3. Taylor expanded in b around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 32.7

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

5. Applied rewrites 32.7%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
6. Add Preprocessing

Alternative 15: 15.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ b a) -0.3333333333333333))

C

double code(double a, double b, double c) {
	return (b / a) * -0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (b / a) * (-0.3333333333333333d0)
end function

Java

public static double code(double a, double b, double c) {
	return (b / a) * -0.3333333333333333;
}

Python

def code(a, b, c):
	return (b / a) * -0.3333333333333333

Julia

function code(a, b, c)
	return Float64(Float64(b / a) * -0.3333333333333333)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (b / a) * -0.3333333333333333;
end

Wolfram

code[a_, b_, c_] := N[(N[(b / a), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 46.9%

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

3. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. sqrt-unprod N/A

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

   5. lower-fma.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. metadata-eval N/A

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

   8. sqrt-unprod N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 19.1

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

5. Applied rewrites 19.1%

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

6. Taylor expanded in b around inf

   \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{b}{a}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{b}{a} \cdot \frac{-1}{3} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{b}{a} \cdot \frac{-1}{3} \]

   3. lift-*.f64 12.8

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

8. Applied rewrites 12.8%

   \[\leadsto \frac{b}{a} \cdot \color{blue}{-0.3333333333333333} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))