Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.7% → 99.0%

Time: 3.4s

Alternatives: 12

Speedup: 3.4×

• 60.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 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) * (1.0 - (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) * (1.0d0 - (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) * (1.0 - (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) * (1.0 - (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(1.0 - 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) * (1.0 - (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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathsf{fma}\left(t\_0, t\_0, \left(b \cdot b\right) \cdot 4 - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a)))) (fma t_0 t_0 (- (* (* b b) 4.0) 1.0))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	return fma(t_0, t_0, (((b * b) * 4.0) - 1.0));
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	return fma(t_0, t_0, Float64(Float64(Float64(b * b) * 4.0) - 1.0))
end

Wolfram

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

Derivation

1. Initial program 73.7%

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

2. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 99.0

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

4. Applied rewrites 99.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate--l+ N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

6. Applied rewrites 99.0%

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

Alternative 2: 94.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+20}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 105000:\\ \;\;\;\;\mathsf{fma}\left(b, b, 4\right) \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e+20)
   (pow a 4.0)
   (if (<= a 105000.0)
     (- (* (fma b b 4.0) (* b b)) 1.0)
     (fma (* 4.0 (* a a)) (+ 1.0 a) (- (* (* a a) (* a a)) 1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e+20) {
		tmp = pow(a, 4.0);
	} else if (a <= 105000.0) {
		tmp = (fma(b, b, 4.0) * (b * b)) - 1.0;
	} else {
		tmp = fma((4.0 * (a * a)), (1.0 + a), (((a * a) * (a * a)) - 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e+20)
		tmp = a ^ 4.0;
	elseif (a <= 105000.0)
		tmp = Float64(Float64(fma(b, b, 4.0) * Float64(b * b)) - 1.0);
	else
		tmp = fma(Float64(4.0 * Float64(a * a)), Float64(1.0 + a), Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e+20], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 105000.0], N[(N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 + a), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.7e20

   1. Initial program 25.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 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. lift-*.f64 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 93.4

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

   6. Applied rewrites 93.4%

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

   if -2.7e20 < a < 105000

   1. Initial program 99.7%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 97.2

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

   4. Applied rewrites 97.2%

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

   6. Applied rewrites 97.2%

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

   if 105000 < a

   1. Initial program 63.7%

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

   2. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{4} - 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{4} - 1\right) \]

      7. lift-+.f64 N/A

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

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{4} - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{\left(2 + 2\right)} - 1\right) \]

      10. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{2} \cdot {a}^{2} - 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{2} \cdot {a}^{2} - 1\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, \left(a \cdot a\right) \cdot {a}^{2} - 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, \left(a \cdot a\right) \cdot {a}^{2} - 1\right) \]

      14. pow2 N/A

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

      15. lift-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

Alternative 3: 94.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+20}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 4\right) \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e+20)
   (pow a 4.0)
   (if (<= a 9.6e+27) (- (* (fma b b 4.0) (* b b)) 1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e+20) {
		tmp = pow(a, 4.0);
	} else if (a <= 9.6e+27) {
		tmp = (fma(b, b, 4.0) * (b * b)) - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e+20)
		tmp = a ^ 4.0;
	elseif (a <= 9.6e+27)
		tmp = Float64(Float64(fma(b, b, 4.0) * Float64(b * b)) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e+20], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 9.6e+27], N[(N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.7e20 or 9.59999999999999991e27 < a

   1. Initial program 44.1%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 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. lift-*.f64 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 93.4

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

   6. Applied rewrites 93.4%

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

   if -2.7e20 < a < 9.59999999999999991e27

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

   6. Applied rewrites 95.9%

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

Alternative 4: 94.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+20}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 4\right) \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e+20)
   (* (* a a) (* a a))
   (if (<= a 9.6e+27) (- (* (fma b b 4.0) (* b b)) 1.0) (* (* (* a a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e+20) {
		tmp = (a * a) * (a * a);
	} else if (a <= 9.6e+27) {
		tmp = (fma(b, b, 4.0) * (b * b)) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e+20)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 9.6e+27)
		tmp = Float64(Float64(fma(b, b, 4.0) * Float64(b * b)) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e+20], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.6e+27], N[(N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.7e20

   1. Initial program 25.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 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. lift-*.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if -2.7e20 < a < 9.59999999999999991e27

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 95.9

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

   4. Applied rewrites 95.9%

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

   6. Applied rewrites 95.9%

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

   if 9.59999999999999991e27 < a

   1. Initial program 63.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 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. lift-*.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. unpow3 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow3 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 93.4

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

   6. Applied rewrites 93.4%

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

Alternative 5: 94.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+20}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+27}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.7e+20)
   (* (* a a) (* a a))
   (if (<= a 9.6e+27) (- (* (* b b) (* b b)) 1.0) (* (* (* a a) a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.7e+20) {
		tmp = (a * a) * (a * a);
	} else if (a <= 9.6e+27) {
		tmp = ((b * b) * (b * b)) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	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 (a <= (-2.7d+20)) then
        tmp = (a * a) * (a * a)
    else if (a <= 9.6d+27) then
        tmp = ((b * b) * (b * b)) - 1.0d0
    else
        tmp = ((a * a) * a) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.7e+20) {
		tmp = (a * a) * (a * a);
	} else if (a <= 9.6e+27) {
		tmp = ((b * b) * (b * b)) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.7e+20:
		tmp = (a * a) * (a * a)
	elif a <= 9.6e+27:
		tmp = ((b * b) * (b * b)) - 1.0
	else:
		tmp = ((a * a) * a) * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.7e+20)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 9.6e+27)
		tmp = Float64(Float64(Float64(b * b) * Float64(b * b)) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.7e+20)
		tmp = (a * a) * (a * a);
	elseif (a <= 9.6e+27)
		tmp = ((b * b) * (b * b)) - 1.0;
	else
		tmp = ((a * a) * a) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.7e+20], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.6e+27], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.7e20

   1. Initial program 25.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 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. lift-*.f64 93.3

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

   4. Applied rewrites 93.3%

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

   if -2.7e20 < a < 9.59999999999999991e27

   1. Initial program 99.0%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 94.8

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

   4. Applied rewrites 94.8%

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

   if 9.59999999999999991e27 < a

   1. Initial program 63.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 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. lift-*.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. unpow3 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow3 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 93.4

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

   6. Applied rewrites 93.4%

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

Alternative 6: 83.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, \left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, \left(\mathsf{fma}\left(a \cdot a, 2, b \cdot b\right) \cdot b\right) \cdot b\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.3e+15)
   (fma (* 4.0 (* a a)) (+ 1.0 a) (- (* (* a a) (* a a)) 1.0))
   (- (fma (* b b) 4.0 (* (* (fma (* a a) 2.0 (* b b)) b) b)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.3e+15) {
		tmp = fma((4.0 * (a * a)), (1.0 + a), (((a * a) * (a * a)) - 1.0));
	} else {
		tmp = fma((b * b), 4.0, ((fma((a * a), 2.0, (b * b)) * b) * b)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.3e+15)
		tmp = fma(Float64(4.0 * Float64(a * a)), Float64(1.0 + a), Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0));
	else
		tmp = Float64(fma(Float64(b * b), 4.0, Float64(Float64(fma(Float64(a * a), 2.0, Float64(b * b)) * b) * b)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.3e+15], N[(N[(4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(1.0 + a), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0 + N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0 + N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3e15

   1. Initial program 76.7%

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

   2. Taylor expanded in b around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{4} - 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{4} - 1\right) \]

      7. lift-+.f64 N/A

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

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{4} - 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{\left(2 + 2\right)} - 1\right) \]

      10. pow-prod-up N/A

          \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{2} \cdot {a}^{2} - 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, {a}^{2} \cdot {a}^{2} - 1\right) \]

      12. pow2 N/A

          \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, \left(a \cdot a\right) \cdot {a}^{2} - 1\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(4 \cdot \left(a \cdot a\right), 1 + a, \left(a \cdot a\right) \cdot {a}^{2} - 1\right) \]

      14. pow2 N/A

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

      15. lift-*.f64 67.5

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

   4. Applied rewrites 67.5%

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

   if 1.3e15 < b

   1. Initial program 63.7%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 98.4

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

   7. Applied rewrites 98.4%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(a \cdot a, 2, b \cdot b\right) \cdot \left(b \cdot b\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.4%

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

Alternative 7: 82.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e-20) (- (* (* a a) 4.0) 1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e-20) {
		tmp = ((a * a) * 4.0) - 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 * b) <= 5d-20) then
        tmp = ((a * a) * 4.0d0) - 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 * b) <= 5e-20) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e-20:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * b) * b) * b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e-20)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 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 * b) <= 5e-20)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * b) * b) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e-20], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 4.9999999999999999e-20

   1. Initial program 82.7%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      3. pow2 N/A

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

      4. lift-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if 4.9999999999999999e-20 < (*.f64 b b)

   1. Initial program 64.8%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 85.7

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

   7. Applied rewrites 85.7%

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

Alternative 8: 81.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 0.24:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 0.24) (- (* (* a a) 4.0) 1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 0.24) {
		tmp = ((a * a) * 4.0) - 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 * b) <= 0.24d0) then
        tmp = ((a * a) * 4.0d0) - 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 * b) <= 0.24) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 0.24:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 0.24)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 0.24)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 0.24], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 0.23999999999999999

   1. Initial program 82.6%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in b around 0

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      3. pow2 N/A

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

      4. lift-*.f64 76.2

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

   7. Applied rewrites 76.2%

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

   if 0.23999999999999999 < (*.f64 b b)

   1. Initial program 64.3%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 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. lift-*.f64 88.3

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

   4. Applied rewrites 88.3%

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

Alternative 9: 74.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.96:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -2.5e+14) t_0 (if (<= a 0.96) (- (* (* b b) 4.0) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -2.5e+14) {
		tmp = t_0;
	} else if (a <= 0.96) {
		tmp = ((b * b) * 4.0) - 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 <= (-2.5d+14)) then
        tmp = t_0
    else if (a <= 0.96d0) then
        tmp = ((b * b) * 4.0d0) - 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 <= -2.5e+14) {
		tmp = t_0;
	} else if (a <= 0.96) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * a)
	tmp = 0
	if a <= -2.5e+14:
		tmp = t_0
	elif a <= 0.96:
		tmp = ((b * b) * 4.0) - 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 <= -2.5e+14)
		tmp = t_0;
	elseif (a <= 0.96)
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 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 <= -2.5e+14)
		tmp = t_0;
	elseif (a <= 0.96)
		tmp = ((b * b) * 4.0) - 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, -2.5e+14], t$95$0, If[LessEqual[a, 0.96], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5e14 or 0.95999999999999996 < a

   1. Initial program 45.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 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. lift-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

   if -2.5e14 < a < 0.95999999999999996

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto {b}^{2} \cdot 4 - 1 \]

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

Alternative 10: 72.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{+129}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+28}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.2e+129)
   (- (* (* a a) 4.0) 1.0)
   (if (<= a 1.2e+28) (- (* (* b b) 4.0) 1.0) (* (* (* a a) a) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.2e+129) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (a <= 1.2e+28) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = ((a * a) * a) * 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 (a <= (-6.2d+129)) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else if (a <= 1.2d+28) then
        tmp = ((b * b) * 4.0d0) - 1.0d0
    else
        tmp = ((a * a) * a) * 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.2e+129) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (a <= 1.2e+28) {
		tmp = ((b * b) * 4.0) - 1.0;
	} else {
		tmp = ((a * a) * a) * 4.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.2e+129:
		tmp = ((a * a) * 4.0) - 1.0
	elif a <= 1.2e+28:
		tmp = ((b * b) * 4.0) - 1.0
	else:
		tmp = ((a * a) * a) * 4.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.2e+129)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	elseif (a <= 1.2e+28)
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * 4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.2e+129)
		tmp = ((a * a) * 4.0) - 1.0;
	elseif (a <= 1.2e+28)
		tmp = ((b * b) * 4.0) - 1.0;
	else
		tmp = ((a * a) * a) * 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.2e+129], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 1.2e+28], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.1999999999999999e129

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in b around 0

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      3. pow2 N/A

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

      4. lift-*.f64 86.1

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

   7. Applied rewrites 86.1%

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

   if -6.1999999999999999e129 < a < 1.19999999999999991e28

   1. Initial program 95.7%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto {b}^{2} \cdot 4 - 1 \]

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if 1.19999999999999991e28 < a

   1. Initial program 63.0%

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

   2. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. lower-*.f64 N/A

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

      11. unpow3 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

         \[\leadsto {a}^{3} \cdot 4 \]

      2. lower-*.f64 N/A

         \[\leadsto {a}^{3} \cdot 4 \]

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 75.8

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

   10. Applied rewrites 75.8%

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

Alternative 11: 61.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.65 \cdot 10^{+135}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.65e+135) (- (* (* a a) 4.0) 1.0) (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.65e+135) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	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 <= 2.65d+135) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((b * b) * 4.0d0) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.65e+135) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.65e+135:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.65e+135)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * 4.0) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.65e+135)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.65e+135], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.65000000000000008e135

   1. Initial program 75.7%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot 4 - 1 \]

      3. pow2 N/A

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

      4. lift-*.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if 2.65000000000000008e135 < b

   1. Initial program 61.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

         \[\leadsto {b}^{2} \cdot 4 - 1 \]

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 91.5

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

   7. Applied rewrites 91.5%

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

Alternative 12: 52.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \left(a \cdot a\right) \cdot 4 - 1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* a a) 4.0) 1.0))

C

double code(double a, double b) {
	return ((a * a) * 4.0) - 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 = ((a * a) * 4.0d0) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return ((a * a) * 4.0) - 1.0;
}

Python

def code(a, b):
	return ((a * a) * 4.0) - 1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(a * a) * 4.0) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((a * a) * 4.0) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 73.7%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 74.4%

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

5. Taylor expanded in b around 0

   \[\leadsto 4 \cdot \color{blue}{{a}^{2}} - 1 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto {a}^{2} \cdot 4 - 1 \]

   2. lower-*.f64 N/A

      \[\leadsto {a}^{2} \cdot 4 - 1 \]

   3. pow2 N/A

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

   4. lift-*.f64 52.3

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

7. Applied rewrites 52.3%

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

Reproduce

herbie shell --seed 2025122 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))