Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.4% → 99.1%

Time: 4.7s

Alternatives: 8

Speedup: 3.8×

• 61.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 73.4% 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: 99.1% accurate, 3.8× 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 70.6%

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

4. Applied rewrites 72.2%

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

5. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 99.0

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

7. Applied rewrites 99.0%

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

Alternative 2: 93.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+74} \lor \neg \left(a \leq 1.2 \cdot 10^{+71}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), b \cdot b, \left(b \cdot b\right) \cdot 4 - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.1e+74) (not (<= a 1.2e+71)))
   (* (* a a) (* a a))
   (fma (fma b b (* a a)) (* b b) (- (* (* b b) 4.0) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.1e+74) || !(a <= 1.2e+71)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = fma(fma(b, b, (a * a)), (b * b), (((b * b) * 4.0) - 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.1e+74) || !(a <= 1.2e+71))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = fma(fma(b, b, Float64(a * a)), Float64(b * b), Float64(Float64(Float64(b * b) * 4.0) - 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.1e+74], N[Not[LessEqual[a, 1.2e+71]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.0999999999999999e74 or 1.1999999999999999e71 < a

   1. Initial program 32.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -2.0999999999999999e74 < a < 1.1999999999999999e71

   1. Initial program 94.1%

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 91.7

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

   10. Applied rewrites 91.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.1 \cdot 10^{+74} \lor \neg \left(a \leq 1.2 \cdot 10^{+71}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), b \cdot b, \left(b \cdot b\right) \cdot 4 - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a)))) (fma t_0 t_0 (- (* (* a a) 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, (((a * a) * 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(a * a) * 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[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 70.6%

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

4. Applied rewrites 72.2%

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

5. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in b around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-+.f64 83.0

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

10. Applied rewrites 83.0%

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

11. Taylor expanded in a around 0

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

    1. pow2 N/A

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

    2. lift-*.f64 98.2

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

13. Applied rewrites 98.2%

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

Alternative 4: 58.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1500000000:\\ \;\;\;\;\left(\left(\left(\frac{4}{a} + 1\right) \cdot \left(a \cdot a\right)\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1500000000.0)
   (* (* (* (+ (/ 4.0 a) 1.0) (* a a)) a) a)
   (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1500000000.0) {
		tmp = ((((4.0 / a) + 1.0) * (a * a)) * a) * a;
	} 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 <= 1500000000.0d0) then
        tmp = ((((4.0d0 / a) + 1.0d0) * (a * a)) * a) * a
    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 <= 1500000000.0) {
		tmp = ((((4.0 / a) + 1.0) * (a * a)) * a) * a;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1500000000.0:
		tmp = ((((4.0 / a) + 1.0) * (a * a)) * a) * a
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1500000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(4.0 / a) + 1.0) * Float64(a * a)) * a) * a);
	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 <= 1500000000.0)
		tmp = ((((4.0 / a) + 1.0) * (a * a)) * a) * a;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1500000000.0], N[(N[(N[(N[(N[(4.0 / a), $MachinePrecision] + 1.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e9

   1. Initial program 76.8%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   4. 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. lower-pow.f64 44.7

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

   5. Applied rewrites 44.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 44.7

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

   7. Applied rewrites 44.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 44.7

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

   9. Applied rewrites 44.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-+.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. associate-*r* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lift-/.f64 N/A

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

       11. lift-+.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 44.7

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

   11. Applied rewrites 44.7%

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

   if 1.5e9 < b

   1. Initial program 55.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 91.1

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

   5. Applied rewrites 91.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

Alternative 5: 58.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1500000000:\\ \;\;\;\;\left(\left(4 + a\right) \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1500000000.0) (* (* (+ 4.0 a) a) (* a a)) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1500000000.0) {
		tmp = ((4.0 + a) * a) * (a * a);
	} 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 <= 1500000000.0d0) then
        tmp = ((4.0d0 + a) * a) * (a * a)
    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 <= 1500000000.0) {
		tmp = ((4.0 + a) * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1500000000.0:
		tmp = ((4.0 + a) * a) * (a * a)
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1500000000.0)
		tmp = Float64(Float64(Float64(4.0 + a) * a) * Float64(a * a));
	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 <= 1500000000.0)
		tmp = ((4.0 + a) * a) * (a * a);
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1500000000.0], N[(N[(N[(4.0 + a), $MachinePrecision] * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e9

   1. Initial program 76.8%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
   4. 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. lower-pow.f64 44.7

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

   5. Applied rewrites 44.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 44.7

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

   7. Applied rewrites 44.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 44.7

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

   9. Applied rewrites 44.7%

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

   10. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 44.7

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

   12. Applied rewrites 44.7%

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

   if 1.5e9 < b

   1. Initial program 55.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 91.1

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

   5. Applied rewrites 91.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

Alternative 6: 57.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1500000000:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1500000000.0) (* (* a a) (* a a)) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1500000000.0) {
		tmp = (a * a) * (a * a);
	} 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 <= 1500000000.0d0) then
        tmp = (a * a) * (a * a)
    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 <= 1500000000.0) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1500000000.0:
		tmp = (a * a) * (a * a)
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1500000000.0)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	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 <= 1500000000.0)
		tmp = (a * a) * (a * a);
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1500000000.0], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e9

   1. Initial program 76.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 44.0

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

   5. Applied rewrites 44.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 44.0

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

   7. Applied rewrites 44.0%

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

   if 1.5e9 < b

   1. Initial program 55.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 91.1

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

   5. Applied rewrites 91.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

Alternative 7: 45.1% accurate, 10.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (* (* a a) (* a a)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(a, b):
	return (a * a) * (a * a)

Julia

function code(a, b)
	return Float64(Float64(a * a) * Float64(a * a))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in a around inf

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

   1. lower-pow.f64 44.3

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

5. Applied rewrites 44.3%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 44.3

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

7. Applied rewrites 44.3%

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

Alternative 8: 18.8% accurate, 10.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (4.0 * a) * (a * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (4.0d0 * a) * (a * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{{a}^{4} \cdot \left(1 + 4 \cdot \frac{1}{a}\right)} \]
4. 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. lower-pow.f64 44.8

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

5. Applied rewrites 44.8%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 44.8

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

7. Applied rewrites 44.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-+.f64 44.8

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

9. Applied rewrites 44.8%

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

10. Taylor expanded in a around 0

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

    1. lower-*.f64 19.4

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

12. Applied rewrites 19.4%

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

Reproduce

herbie shell --seed 2025075 
(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))