Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.6% → 99.7%

Time: 3.2s

Alternatives: 10

Speedup: 3.3×

• 60.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (3.0d0 + a))))) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(\left(1 - a\right) \cdot a, a, \left(\left(3 + a\right) \cdot b\right) \cdot b\right), 4, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(3 \cdot b, b, a \cdot a\right), -4, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<=
        (-
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
         1.0)
        INFINITY)
     (fma t_0 t_0 (fma (fma (* (- 1.0 a) a) a (* (* (+ 3.0 a) b) b)) 4.0 -1.0))
     (- (* (* (* a a) a) a) (fma (fma (* 3.0 b) b (* a a)) -4.0 1.0)))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0) <= ((double) INFINITY)) {
		tmp = fma(t_0, t_0, fma(fma(((1.0 - a) * a), a, (((3.0 + a) * b) * b)), 4.0, -1.0));
	} else {
		tmp = (((a * a) * a) * a) - fma(fma((3.0 * b), b, (a * a)), -4.0, 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + Float64(Float64(b * b) * Float64(3.0 + a))))) - 1.0) <= Inf)
		tmp = fma(t_0, t_0, fma(fma(Float64(Float64(1.0 - a) * a), a, Float64(Float64(Float64(3.0 + a) * b) * b)), 4.0, -1.0));
	else
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) - fma(fma(Float64(3.0 * b), b, Float64(a * a)), -4.0, 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], Infinity], N[(t$95$0 * t$95$0 + N[(N[(N[(N[(1.0 - a), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(3.0 + a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - N[(N[(N[(3.0 * b), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * -4.0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a + b \cdot b, a \cdot a + b \cdot b, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot a + b \cdot b}, a \cdot a + b \cdot b, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b + a \cdot a}, a \cdot a + b \cdot b, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b} + a \cdot a, a \cdot a + b \cdot b, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, b, a \cdot a\right)}, a \cdot a + b \cdot b, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \color{blue}{a \cdot a + b \cdot b}, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \color{blue}{b \cdot b + a \cdot a}, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \color{blue}{b \cdot b} + a \cdot a, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \color{blue}{\mathsf{fma}\left(b, b, a \cdot a\right)}, 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - 1\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b, b, a \cdot a\right), 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right) - \color{blue}{1 \cdot 1}\right) \]

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(b, b, a \cdot a\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(1 - a\right) \cdot a, a, \left(\left(3 + a\right) \cdot b\right) \cdot b\right), 4, -1\right)\right)} \]

   if +inf.0 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 0.0%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

   6. Applied rewrites 5.2%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \left(\left(1 - a\right) \cdot a\right) \cdot a\right), -4, 1\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \color{blue}{a} \cdot a\right), -4, 1\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.3%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \color{blue}{a} \cdot a\right), -4, 1\right) \]

   10. Taylor expanded in a around 0

       \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{3} \cdot b, b, a \cdot a\right), -4, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 99.3%

       \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{3} \cdot b, b, a \cdot a\right), -4, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(3 \cdot b, b, a \cdot a\right), -4, 1\right)\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* (* (* a a) a) a) (fma (fma (* 3.0 b) b (* a a)) -4.0 1.0))))
   (if (<= a -4.1e+14)
     t_0
     (if (<= a 1e-16) (- (fma 12.0 (pow b 2.0) (pow b 4.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (((a * a) * a) * a) - fma(fma((3.0 * b), b, (a * a)), -4.0, 1.0);
	double tmp;
	if (a <= -4.1e+14) {
		tmp = t_0;
	} else if (a <= 1e-16) {
		tmp = fma(12.0, pow(b, 2.0), pow(b, 4.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(Float64(a * a) * a) * a) - fma(fma(Float64(3.0 * b), b, Float64(a * a)), -4.0, 1.0))
	tmp = 0.0
	if (a <= -4.1e+14)
		tmp = t_0;
	elseif (a <= 1e-16)
		tmp = Float64(fma(12.0, (b ^ 2.0), (b ^ 4.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - N[(N[(N[(3.0 * b), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * -4.0 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.1e+14], t$95$0, If[LessEqual[a, 1e-16], N[(N[(12.0 * N[Power[b, 2.0], $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1e14 or 9.9999999999999998e-17 < a

   1. Initial program 49.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.2

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

   4. Applied rewrites 45.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

   6. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \left(\left(1 - a\right) \cdot a\right) \cdot a\right), -4, 1\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \color{blue}{a} \cdot a\right), -4, 1\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 77.9%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \color{blue}{a} \cdot a\right), -4, 1\right) \]

   10. Taylor expanded in a around 0

       \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{3} \cdot b, b, a \cdot a\right), -4, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 93.8%

       \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{3} \cdot b, b, a \cdot a\right), -4, 1\right) \]

   if -4.1e14 < a < 9.9999999999999998e-17

   1. Initial program 99.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 98.2

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(3 \cdot b, b, a \cdot a\right), -4, 1\right)\\ \mathbf{if}\;a \leq -4.1 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 10^{-16}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 12\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* (* (* a a) a) a) (fma (fma (* 3.0 b) b (* a a)) -4.0 1.0))))
   (if (<= a -4.1e+14)
     t_0
     (if (<= a 1e-16) (- (* (* b b) (fma b b 12.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (((a * a) * a) * a) - fma(fma((3.0 * b), b, (a * a)), -4.0, 1.0);
	double tmp;
	if (a <= -4.1e+14) {
		tmp = t_0;
	} else if (a <= 1e-16) {
		tmp = ((b * b) * fma(b, b, 12.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(Float64(a * a) * a) * a) - fma(fma(Float64(3.0 * b), b, Float64(a * a)), -4.0, 1.0))
	tmp = 0.0
	if (a <= -4.1e+14)
		tmp = t_0;
	elseif (a <= 1e-16)
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 12.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - N[(N[(N[(3.0 * b), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * -4.0 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.1e+14], t$95$0, If[LessEqual[a, 1e-16], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.1e14 or 9.9999999999999998e-17 < a

   1. Initial program 49.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.2

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

   4. Applied rewrites 45.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

   6. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \left(\left(1 - a\right) \cdot a\right) \cdot a\right), -4, 1\right)} \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \color{blue}{a} \cdot a\right), -4, 1\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 77.9%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \color{blue}{a} \cdot a\right), -4, 1\right) \]

   10. Taylor expanded in a around 0

       \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{3} \cdot b, b, a \cdot a\right), -4, 1\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 93.8%

       \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{3} \cdot b, b, a \cdot a\right), -4, 1\right) \]

   if -4.1e14 < a < 9.9999999999999998e-17

   1. Initial program 99.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 98.2

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({b}^{4} + \color{blue}{12 \cdot {b}^{2}}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + 12 \cdot {b}^{\color{blue}{2}}\right) - 1 \]

      4. pow2 N/A

         \[\leadsto \left({b}^{4} + 12 \cdot \left(b \cdot \color{blue}{b}\right)\right) - 1 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

      6. metadata-eval N/A

         \[\leadsto \left({b}^{\left(2 + 2\right)} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      7. pow-prod-up N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot {b}^{2} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      9. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      10. distribute-rgt-out N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{b \cdot b} + 12\right) - 1 \]

      13. lower-fma.f64 98.1

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, \color{blue}{b}, 12\right) - 1 \]

   6. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 12\right)} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;\left(a \cdot \left(a - 4\right)\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.6e+24) (- (* (* a (- a 4.0)) (* a a)) 1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.6e+24) {
		tmp = ((a * (a - 4.0)) * (a * a)) - 1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 5.6d+24) then
        tmp = ((a * (a - 4.0d0)) * (a * a)) - 1.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.6e+24) {
		tmp = ((a * (a - 4.0)) * (a * a)) - 1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.6e+24:
		tmp = ((a * (a - 4.0)) * (a * a)) - 1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.6e+24)
		tmp = Float64(Float64(Float64(a * Float64(a - 4.0)) * Float64(a * a)) - 1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.6e+24)
		tmp = ((a * (a - 4.0)) * (a * a)) - 1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.6e+24], N[(N[(N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.6000000000000003e24

   1. Initial program 77.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 - 4 \cdot \frac{1}{a}\right)} - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(\color{blue}{1} - 4 \cdot \frac{1}{a}\right) - 1 \]

      3. lower--.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 - \color{blue}{4 \cdot \frac{1}{a}}\right) - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 - 4 \cdot \color{blue}{\frac{1}{a}}\right) - 1 \]

      5. lower-/.f64 77.6

         \[\leadsto {a}^{4} \cdot \left(1 - 4 \cdot \frac{1}{\color{blue}{a}}\right) - 1 \]

   4. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 - 4 \cdot \frac{1}{a}\right)} - 1 \]

      2. *-commutative N/A

         \[\leadsto \left(1 - 4 \cdot \frac{1}{a}\right) \cdot \color{blue}{{a}^{4}} - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(1 - 4 \cdot \frac{1}{a}\right) \cdot {a}^{\color{blue}{4}} - 1 \]

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

         \[\leadsto \left(1 - 4 \cdot \frac{1}{a}\right) \cdot {\left(a \cdot a\right)}^{2} - 1 \]

      8. pow2 N/A

         \[\leadsto \left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) - 1 \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} - 1 \]

      11. lower-*.f64 77.5

          \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      14. associate-*r/ N/A

          \[\leadsto \left(\left(1 - \frac{4 \cdot 1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(1 - \frac{4}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      16. lower-/.f64 77.5

          \[\leadsto \left(\left(1 - \frac{4}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

   6. Applied rewrites 77.5%

      \[\leadsto \left(\left(1 - \frac{4}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} - 1 \]

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(a - 4\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      2. lower--.f64 77.6

         \[\leadsto \left(a \cdot \left(a - 4\right)\right) \cdot \left(a \cdot a\right) - 1 \]

   9. Applied rewrites 77.6%

      \[\leadsto \left(a \cdot \left(a - 4\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) - 1 \]

   if 5.6000000000000003e24 < b

   1. Initial program 64.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 92.4

         \[\leadsto {b}^{\color{blue}{4}} \]

   4. Applied rewrites 92.4%

      \[\leadsto \color{blue}{{b}^{4}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;\left(a \cdot \left(a - 4\right)\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.6e+24) (- (* (* a (- a 4.0)) (* a a)) 1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.6e+24) {
		tmp = ((a * (a - 4.0)) * (a * a)) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 5.6d+24) then
        tmp = ((a * (a - 4.0d0)) * (a * a)) - 1.0d0
    else
        tmp = ((b * b) * b) * b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.6e+24) {
		tmp = ((a * (a - 4.0)) * (a * a)) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.6e+24:
		tmp = ((a * (a - 4.0)) * (a * a)) - 1.0
	else:
		tmp = ((b * b) * b) * b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.6e+24)
		tmp = Float64(Float64(Float64(a * Float64(a - 4.0)) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.6e+24)
		tmp = ((a * (a - 4.0)) * (a * a)) - 1.0;
	else
		tmp = ((b * b) * b) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.6e+24], N[(N[(N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.6000000000000003e24

   1. Initial program 77.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 - 4 \cdot \frac{1}{a}\right)} - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(\color{blue}{1} - 4 \cdot \frac{1}{a}\right) - 1 \]

      3. lower--.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 - \color{blue}{4 \cdot \frac{1}{a}}\right) - 1 \]

      4. lower-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \left(1 - 4 \cdot \color{blue}{\frac{1}{a}}\right) - 1 \]

      5. lower-/.f64 77.6

         \[\leadsto {a}^{4} \cdot \left(1 - 4 \cdot \frac{1}{\color{blue}{a}}\right) - 1 \]

   4. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

         \[\leadsto {a}^{4} \cdot \color{blue}{\left(1 - 4 \cdot \frac{1}{a}\right)} - 1 \]

      2. *-commutative N/A

         \[\leadsto \left(1 - 4 \cdot \frac{1}{a}\right) \cdot \color{blue}{{a}^{4}} - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(1 - 4 \cdot \frac{1}{a}\right) \cdot {a}^{\color{blue}{4}} - 1 \]

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

         \[\leadsto \left(1 - 4 \cdot \frac{1}{a}\right) \cdot {\left(a \cdot a\right)}^{2} - 1 \]

      8. pow2 N/A

         \[\leadsto \left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) - 1 \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} - 1 \]

      11. lower-*.f64 77.5

          \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      13. lift-/.f64 N/A

          \[\leadsto \left(\left(1 - 4 \cdot \frac{1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      14. associate-*r/ N/A

          \[\leadsto \left(\left(1 - \frac{4 \cdot 1}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(1 - \frac{4}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      16. lower-/.f64 77.5

          \[\leadsto \left(\left(1 - \frac{4}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot a\right) - 1 \]

   6. Applied rewrites 77.5%

      \[\leadsto \left(\left(1 - \frac{4}{a}\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)} - 1 \]

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(a - 4\right)\right) \cdot \left(a \cdot a\right) - 1 \]

      2. lower--.f64 77.6

         \[\leadsto \left(a \cdot \left(a - 4\right)\right) \cdot \left(a \cdot a\right) - 1 \]

   9. Applied rewrites 77.6%

      \[\leadsto \left(a \cdot \left(a - 4\right)\right) \cdot \left(\color{blue}{a} \cdot a\right) - 1 \]

   if 5.6000000000000003e24 < b

   1. Initial program 64.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 38.4

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

   4. Applied rewrites 38.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

   6. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \left(\left(1 - a\right) \cdot a\right) \cdot a\right), -4, 1\right)} \]

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   8. Step-by-step derivation

      1. lower-pow.f64 92.4

         \[\leadsto {b}^{\color{blue}{4}} \]

   9. Applied rewrites 92.4%

      \[\leadsto \color{blue}{{b}^{4}} \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto {b}^{\color{blue}{4}} \]

       2. metadata-eval N/A

          \[\leadsto {b}^{\left(2 + \color{blue}{2}\right)} \]

       3. pow-prod-up N/A

          \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

       4. pow2 N/A

          \[\leadsto \left(b \cdot b\right) \cdot {\color{blue}{b}}^{2} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot {\color{blue}{b}}^{2} \]

       6. pow2 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

       7. associate-*r* N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \color{blue}{b} \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \color{blue}{b} \]

       9. lower-*.f64 92.4

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot b \]

   11. Applied rewrites 92.4%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.6e+24) (- (* (* (* a a) a) a) 1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.6e+24) {
		tmp = (((a * a) * a) * a) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 5.6d+24) then
        tmp = (((a * a) * a) * a) - 1.0d0
    else
        tmp = ((b * b) * b) * b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.6e+24) {
		tmp = (((a * a) * a) * a) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.6e+24:
		tmp = (((a * a) * a) * a) - 1.0
	else:
		tmp = ((b * b) * b) * b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.6e+24)
		tmp = Float64(Float64(Float64(Float64(a * a) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.6e+24)
		tmp = (((a * a) * a) * a) - 1.0;
	else
		tmp = ((b * b) * b) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.6e+24], N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.6000000000000003e24

   1. Initial program 77.7%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   3. Step-by-step derivation

      1. lower-pow.f64 77.2

         \[\leadsto {a}^{\color{blue}{4}} - 1 \]

   4. Applied rewrites 77.2%

      \[\leadsto \color{blue}{{a}^{4}} - 1 \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} - 1 \]

      2. metadata-eval N/A

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

      3. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} - 1 \]

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]

      9. unpow3 N/A

         \[\leadsto {a}^{3} \cdot a - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto {a}^{3} \cdot \color{blue}{a} - 1 \]

      11. unpow3 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]

      13. lower-*.f64 77.2

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a - 1 \]

   6. Applied rewrites 77.2%

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

   if 5.6000000000000003e24 < b

   1. Initial program 64.4%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 38.4

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

   4. Applied rewrites 38.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

   6. Applied rewrites 41.4%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \left(\left(1 - a\right) \cdot a\right) \cdot a\right), -4, 1\right)} \]

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   8. Step-by-step derivation

      1. lower-pow.f64 92.4

         \[\leadsto {b}^{\color{blue}{4}} \]

   9. Applied rewrites 92.4%

      \[\leadsto \color{blue}{{b}^{4}} \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto {b}^{\color{blue}{4}} \]

       2. metadata-eval N/A

          \[\leadsto {b}^{\left(2 + \color{blue}{2}\right)} \]

       3. pow-prod-up N/A

          \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

       4. pow2 N/A

          \[\leadsto \left(b \cdot b\right) \cdot {\color{blue}{b}}^{2} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot {\color{blue}{b}}^{2} \]

       6. pow2 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

       7. associate-*r* N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \color{blue}{b} \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \color{blue}{b} \]

       9. lower-*.f64 92.4

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot b \]

   11. Applied rewrites 92.4%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-28}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \mathbf{elif}\;a \leq 2.85:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* (* a a) a) a)))
   (if (<= a -2.9e+27)
     t_0
     (if (<= a -1.1e-28)
       (* (* (* b b) b) b)
       (if (<= a 2.85) (- (* (* b b) 12.0) 1.0) t_0)))))

C

double code(double a, double b) {
	double t_0 = ((a * a) * a) * a;
	double tmp;
	if (a <= -2.9e+27) {
		tmp = t_0;
	} else if (a <= -1.1e-28) {
		tmp = ((b * b) * b) * b;
	} else if (a <= 2.85) {
		tmp = ((b * b) * 12.0) - 1.0;
	} else {
		tmp = t_0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((a * a) * a) * a
    if (a <= (-2.9d+27)) then
        tmp = t_0
    else if (a <= (-1.1d-28)) then
        tmp = ((b * b) * b) * b
    else if (a <= 2.85d0) then
        tmp = ((b * b) * 12.0d0) - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = ((a * a) * a) * a;
	double tmp;
	if (a <= -2.9e+27) {
		tmp = t_0;
	} else if (a <= -1.1e-28) {
		tmp = ((b * b) * b) * b;
	} else if (a <= 2.85) {
		tmp = ((b * b) * 12.0) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((a * a) * a) * a
	tmp = 0
	if a <= -2.9e+27:
		tmp = t_0
	elif a <= -1.1e-28:
		tmp = ((b * b) * b) * b
	elif a <= 2.85:
		tmp = ((b * b) * 12.0) - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * a) * a)
	tmp = 0.0
	if (a <= -2.9e+27)
		tmp = t_0;
	elseif (a <= -1.1e-28)
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	elseif (a <= 2.85)
		tmp = Float64(Float64(Float64(b * b) * 12.0) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((a * a) * a) * a;
	tmp = 0.0;
	if (a <= -2.9e+27)
		tmp = t_0;
	elseif (a <= -1.1e-28)
		tmp = ((b * b) * b) * b;
	elseif (a <= 2.85)
		tmp = ((b * b) * 12.0) - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -2.9e+27], t$95$0, If[LessEqual[a, -1.1e-28], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[a, 2.85], N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.9000000000000001e27 or 2.85000000000000009 < a

   1. Initial program 47.5%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 90.0

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 90.0%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 \cdot \color{blue}{2}\right)} \]

      3. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      6. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

      9. unpow3 N/A

         \[\leadsto {a}^{3} \cdot a \]

      10. lower-*.f64 N/A

          \[\leadsto {a}^{3} \cdot \color{blue}{a} \]

      11. unpow3 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

      13. lower-*.f64 90.0

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

   6. Applied rewrites 90.0%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

   if -2.9000000000000001e27 < a < -1.09999999999999998e-28

   1. Initial program 88.3%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 65.2

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

   4. Applied rewrites 65.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate--l- N/A

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

      6. lower--.f64 N/A

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

   6. Applied rewrites 65.1%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a - \mathsf{fma}\left(\mathsf{fma}\left(\left(3 + a\right) \cdot b, b, \left(\left(1 - a\right) \cdot a\right) \cdot a\right), -4, 1\right)} \]

   7. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{4}} \]
   8. Step-by-step derivation

      1. lower-pow.f64 48.7

         \[\leadsto {b}^{\color{blue}{4}} \]

   9. Applied rewrites 48.7%

      \[\leadsto \color{blue}{{b}^{4}} \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto {b}^{\color{blue}{4}} \]

       2. metadata-eval N/A

          \[\leadsto {b}^{\left(2 + \color{blue}{2}\right)} \]

       3. pow-prod-up N/A

          \[\leadsto {b}^{2} \cdot \color{blue}{{b}^{2}} \]

       4. pow2 N/A

          \[\leadsto \left(b \cdot b\right) \cdot {\color{blue}{b}}^{2} \]

       5. lift-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot {\color{blue}{b}}^{2} \]

       6. pow2 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

       7. associate-*r* N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \color{blue}{b} \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \color{blue}{b} \]

       9. lower-*.f64 48.6

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot b \]

   11. Applied rewrites 48.6%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \color{blue}{b} \]

   if -1.09999999999999998e-28 < a < 2.85000000000000009

   1. Initial program 99.9%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 99.5

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({b}^{4} + \color{blue}{12 \cdot {b}^{2}}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + 12 \cdot {b}^{\color{blue}{2}}\right) - 1 \]

      4. pow2 N/A

         \[\leadsto \left({b}^{4} + 12 \cdot \left(b \cdot \color{blue}{b}\right)\right) - 1 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

      6. metadata-eval N/A

         \[\leadsto \left({b}^{\left(2 + 2\right)} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      7. pow-prod-up N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot {b}^{2} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      9. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      10. distribute-rgt-out N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{b \cdot b} + 12\right) - 1 \]

      13. lower-fma.f64 99.4

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, \color{blue}{b}, 12\right) - 1 \]

   6. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 12\right)} - 1 \]

   7. Taylor expanded in b around 0

      \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 76.3%

      \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.85:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* (* a a) a) a)))
   (if (<= a -2.3e+26) t_0 (if (<= a 2.85) (- (* (* b b) 12.0) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = ((a * a) * a) * a;
	double tmp;
	if (a <= -2.3e+26) {
		tmp = t_0;
	} else if (a <= 2.85) {
		tmp = ((b * b) * 12.0) - 1.0;
	} else {
		tmp = t_0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((a * a) * a) * a
    if (a <= (-2.3d+26)) then
        tmp = t_0
    else if (a <= 2.85d0) then
        tmp = ((b * b) * 12.0d0) - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = ((a * a) * a) * a;
	double tmp;
	if (a <= -2.3e+26) {
		tmp = t_0;
	} else if (a <= 2.85) {
		tmp = ((b * b) * 12.0) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((a * a) * a) * a
	tmp = 0
	if a <= -2.3e+26:
		tmp = t_0
	elif a <= 2.85:
		tmp = ((b * b) * 12.0) - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * a) * a)
	tmp = 0.0
	if (a <= -2.3e+26)
		tmp = t_0;
	elseif (a <= 2.85)
		tmp = Float64(Float64(Float64(b * b) * 12.0) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((a * a) * a) * a;
	tmp = 0.0;
	if (a <= -2.3e+26)
		tmp = t_0;
	elseif (a <= 2.85)
		tmp = ((b * b) * 12.0) - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -2.3e+26], t$95$0, If[LessEqual[a, 2.85], N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3000000000000001e26 or 2.85000000000000009 < a

   1. Initial program 47.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 89.9

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 89.9%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 \cdot \color{blue}{2}\right)} \]

      3. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot {\color{blue}{a}}^{2} \]

      6. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(a \cdot \color{blue}{a}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

      9. unpow3 N/A

         \[\leadsto {a}^{3} \cdot a \]

      10. lower-*.f64 N/A

          \[\leadsto {a}^{3} \cdot \color{blue}{a} \]

      11. unpow3 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

      13. lower-*.f64 89.9

          \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot a \]

   6. Applied rewrites 89.9%

      \[\leadsto \left(\left(a \cdot a\right) \cdot a\right) \cdot \color{blue}{a} \]

   if -2.3000000000000001e26 < a < 2.85000000000000009

   1. Initial program 98.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 96.8

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({b}^{4} + \color{blue}{12 \cdot {b}^{2}}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + 12 \cdot {b}^{\color{blue}{2}}\right) - 1 \]

      4. pow2 N/A

         \[\leadsto \left({b}^{4} + 12 \cdot \left(b \cdot \color{blue}{b}\right)\right) - 1 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

      6. metadata-eval N/A

         \[\leadsto \left({b}^{\left(2 + 2\right)} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      7. pow-prod-up N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot {b}^{2} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      9. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      10. distribute-rgt-out N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{b \cdot b} + 12\right) - 1 \]

      13. lower-fma.f64 96.7

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, \color{blue}{b}, 12\right) - 1 \]

   6. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 12\right)} - 1 \]

   7. Taylor expanded in b around 0

      \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 73.7%

      \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 80.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 2.85:\\ \;\;\;\;\left(b \cdot b\right) \cdot 12 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -2.3e+26) t_0 (if (<= a 2.85) (- (* (* b b) 12.0) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -2.3e+26) {
		tmp = t_0;
	} else if (a <= 2.85) {
		tmp = ((b * b) * 12.0) - 1.0;
	} else {
		tmp = t_0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (a * a)
    if (a <= (-2.3d+26)) then
        tmp = t_0
    else if (a <= 2.85d0) then
        tmp = ((b * b) * 12.0d0) - 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -2.3e+26) {
		tmp = t_0;
	} else if (a <= 2.85) {
		tmp = ((b * b) * 12.0) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * a)
	tmp = 0
	if a <= -2.3e+26:
		tmp = t_0
	elif a <= 2.85:
		tmp = ((b * b) * 12.0) - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -2.3e+26)
		tmp = t_0;
	elseif (a <= 2.85)
		tmp = Float64(Float64(Float64(b * b) * 12.0) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * (a * a);
	tmp = 0.0;
	if (a <= -2.3e+26)
		tmp = t_0;
	elseif (a <= 2.85)
		tmp = ((b * b) * 12.0) - 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.3e+26], t$95$0, If[LessEqual[a, 2.85], N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3000000000000001e26 or 2.85000000000000009 < a

   1. Initial program 47.6%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4}} \]
   3. Step-by-step derivation

      1. lower-pow.f64 89.9

         \[\leadsto {a}^{\color{blue}{4}} \]

   4. Applied rewrites 89.9%

      \[\leadsto \color{blue}{{a}^{4}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {a}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto {a}^{\left(2 \cdot \color{blue}{2}\right)} \]

      3. pow-sqr N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{{a}^{2}} \]

      4. pow-prod-down N/A

         \[\leadsto {\left(a \cdot a\right)}^{\color{blue}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot a\right)}^{2} \]

      6. pow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

      7. lower-*.f64 89.8

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   6. Applied rewrites 89.8%

      \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)} \]

   if -2.3000000000000001e26 < a < 2.85000000000000009

   1. Initial program 98.8%

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

      2. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

      3. lower-pow.f64 96.8

         \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

   4. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{4}}\right) - 1 \]

      2. +-commutative N/A

         \[\leadsto \left({b}^{4} + \color{blue}{12 \cdot {b}^{2}}\right) - 1 \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + 12 \cdot {b}^{\color{blue}{2}}\right) - 1 \]

      4. pow2 N/A

         \[\leadsto \left({b}^{4} + 12 \cdot \left(b \cdot \color{blue}{b}\right)\right) - 1 \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({b}^{4} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

      6. metadata-eval N/A

         \[\leadsto \left({b}^{\left(2 + 2\right)} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      7. pow-prod-up N/A

         \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

      8. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot {b}^{2} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      9. pow2 N/A

         \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

      10. distribute-rgt-out N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{b \cdot b} + 12\right) - 1 \]

      13. lower-fma.f64 96.7

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, \color{blue}{b}, 12\right) - 1 \]

   6. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 12\right)} - 1 \]

   7. Taylor expanded in b around 0

      \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 73.7%

      \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \left(b \cdot b\right) \cdot 12 - 1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* b b) 12.0) 1.0))

C

double code(double a, double b) {
	return ((b * b) * 12.0) - 1.0;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((b * b) * 12.0d0) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return ((b * b) * 12.0) - 1.0;
}

Python

def code(a, b):
	return ((b * b) * 12.0) - 1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(b * b) * 12.0) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((b * b) * 12.0) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 74.6%

   \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(12 \cdot {b}^{2} + {b}^{4}\right)} - 1 \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(12, \color{blue}{{b}^{2}}, {b}^{4}\right) - 1 \]

   2. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(12, {b}^{\color{blue}{2}}, {b}^{4}\right) - 1 \]

   3. lower-pow.f64 69.3

      \[\leadsto \mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right) - 1 \]

4. Applied rewrites 69.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(12, {b}^{2}, {b}^{4}\right)} - 1 \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \left(12 \cdot {b}^{2} + \color{blue}{{b}^{4}}\right) - 1 \]

   2. +-commutative N/A

      \[\leadsto \left({b}^{4} + \color{blue}{12 \cdot {b}^{2}}\right) - 1 \]

   3. lift-pow.f64 N/A

      \[\leadsto \left({b}^{4} + 12 \cdot {b}^{\color{blue}{2}}\right) - 1 \]

   4. pow2 N/A

      \[\leadsto \left({b}^{4} + 12 \cdot \left(b \cdot \color{blue}{b}\right)\right) - 1 \]

   5. lift-pow.f64 N/A

      \[\leadsto \left({b}^{4} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

   6. metadata-eval N/A

      \[\leadsto \left({b}^{\left(2 + 2\right)} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

   7. pow-prod-up N/A

      \[\leadsto \left({b}^{2} \cdot {b}^{2} + \color{blue}{12} \cdot \left(b \cdot b\right)\right) - 1 \]

   8. pow2 N/A

      \[\leadsto \left(\left(b \cdot b\right) \cdot {b}^{2} + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

   9. pow2 N/A

      \[\leadsto \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + 12 \cdot \left(b \cdot b\right)\right) - 1 \]

   10. distribute-rgt-out N/A

       \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

   11. lower-*.f64 N/A

       \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot b + 12\right)} - 1 \]

   12. lower-*.f64 N/A

       \[\leadsto \left(b \cdot b\right) \cdot \left(\color{blue}{b \cdot b} + 12\right) - 1 \]

   13. lower-fma.f64 69.2

       \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, \color{blue}{b}, 12\right) - 1 \]

6. Applied rewrites 69.2%

   \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 12\right)} - 1 \]

7. Taylor expanded in b around 0

   \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
8. Step-by-step derivation

9. Applied rewrites 51.1%

   \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025113 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))