Bouland and Aaronson, Equation (25)

Percentage Accurate: 74.4% → 99.1%

Time: 3.3s

Alternatives: 10

Speedup: 3.2×

• 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: 74.4% 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, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

3. Applied rewrites 75.9%

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

4. Taylor expanded in a around 0

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

6. Applied rewrites 83.7%

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

7. Taylor expanded in a around 0

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

9. Applied rewrites 99.1%

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

Alternative 2: 94.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{+23}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+52}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.05e+23)
   (pow a 4.0)
   (if (<= a 3.7e+52) (- (* (* (fma b b 4.0) b) b) 1.0) (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.05e+23) {
		tmp = pow(a, 4.0);
	} else if (a <= 3.7e+52) {
		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 <= -1.05e+23)
		tmp = a ^ 4.0;
	elseif (a <= 3.7e+52)
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.05e+23], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 3.7e+52], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.0500000000000001e23 or 3.7e52 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 44.9

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

   4. Applied rewrites 44.9%

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

   if -1.0500000000000001e23 < a < 3.7e52

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 70.0

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

   6. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 70.0

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

   8. Applied rewrites 70.0%

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

Alternative 3: 94.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+52}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* (* a a) a) a)))
   (if (<= a -1.05e+23)
     t_0
     (if (<= a 3.7e+52) (- (* (* (fma b b 4.0) b) b) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = ((a * a) * a) * a;
	double tmp;
	if (a <= -1.05e+23) {
		tmp = t_0;
	} else if (a <= 3.7e+52) {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * a) * a)
	tmp = 0.0
	if (a <= -1.05e+23)
		tmp = t_0;
	elseif (a <= 3.7e+52)
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1.05e+23], t$95$0, If[LessEqual[a, 3.7e+52], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.0500000000000001e23 or 3.7e52 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 44.9

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

   4. Applied rewrites 44.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up 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. pow2 N/A

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

      6. associate-*r* N/A

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

      7. pow3 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 44.9

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

   6. Applied rewrites 44.9%

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

   if -1.0500000000000001e23 < a < 3.7e52

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 70.0

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

   6. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 70.0

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

   8. Applied rewrites 70.0%

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

Alternative 4: 94.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{+52}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(b, b, 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* (* a a) a) a)))
   (if (<= a -1.05e+23)
     t_0
     (if (<= a 3.7e+52) (- (* (* b b) (fma 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 <= -1.05e+23) {
		tmp = t_0;
	} else if (a <= 3.7e+52) {
		tmp = ((b * b) * fma(b, b, 4.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * a) * a)
	tmp = 0.0
	if (a <= -1.05e+23)
		tmp = t_0;
	elseif (a <= 3.7e+52)
		tmp = Float64(Float64(Float64(b * b) * fma(b, b, 4.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1.05e+23], t$95$0, If[LessEqual[a, 3.7e+52], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.0500000000000001e23 or 3.7e52 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 44.9

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

   4. Applied rewrites 44.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up 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. pow2 N/A

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

      6. associate-*r* N/A

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

      7. pow3 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 44.9

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

   6. Applied rewrites 44.9%

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

   if -1.0500000000000001e23 < a < 3.7e52

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 70.0

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

   6. Applied rewrites 70.0%

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

Alternative 5: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 23.5:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, b \cdot b, {b}^{4}\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 23.5)
   (- (fma (* a a) 4.0 (* (* (* a a) a) a)) 1.0)
   (- (fma 4.0 (* b b) (pow b 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 23.5) {
		tmp = fma((a * a), 4.0, (((a * a) * a) * a)) - 1.0;
	} else {
		tmp = fma(4.0, (b * b), pow(b, 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 23.5)
		tmp = Float64(fma(Float64(a * a), 4.0, Float64(Float64(Float64(a * a) * a) * a)) - 1.0);
	else
		tmp = Float64(fma(4.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 23.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0 + N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(4.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 23.5

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 53.2

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

   4. Applied rewrites 53.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 53.2

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

      16. lift-pow.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow-prod-up N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. associate-*r* N/A

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

      22. pow3 N/A

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

      23. lower-*.f64 N/A

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

      24. pow3 N/A

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

      25. lower-*.f64 N/A

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

      26. lift-*.f64 53.1

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

   6. Applied rewrites 53.1%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 68.6%

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

   if 23.5 < b

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 70.1

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

   6. Applied rewrites 70.1%

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

Alternative 6: 81.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1550000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, 4, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1550000000.0)
   (- (fma (* a a) 4.0 (* (* (* a a) a) a)) 1.0)
   (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1550000000.0) {
		tmp = fma((a * a), 4.0, (((a * a) * a) * a)) - 1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1550000000.0)
		tmp = Float64(fma(Float64(a * a), 4.0, Float64(Float64(Float64(a * a) * a) * a)) - 1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1550000000.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0 + N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.55e9

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 53.2

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

   4. Applied rewrites 53.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 53.2

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

      16. lift-pow.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow-prod-up N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. associate-*r* N/A

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

      22. pow3 N/A

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

      23. lower-*.f64 N/A

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

      24. pow3 N/A

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

      25. lower-*.f64 N/A

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

      26. lift-*.f64 53.1

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

   6. Applied rewrites 53.1%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 68.6%

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

   if 1.55e9 < b

   1. Initial program 74.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 75.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 83.7%

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

   7. Taylor expanded in b around inf

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

      1. lower-pow.f64 45.9

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

   9. Applied rewrites 45.9%

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

Alternative 7: 81.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -21000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+33}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* (* a a) a) a)))
   (if (<= a -21000000.0)
     t_0
     (if (<= a 4.8e+33) (- (* (* 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 <= -21000000.0) {
		tmp = t_0;
	} else if (a <= 4.8e+33) {
		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 <= (-21000000.0d0)) then
        tmp = t_0
    else if (a <= 4.8d+33) 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 <= -21000000.0) {
		tmp = t_0;
	} else if (a <= 4.8e+33) {
		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 <= -21000000.0:
		tmp = t_0
	elif a <= 4.8e+33:
		tmp = ((b * b) * 4.0) - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * a) * a)
	tmp = 0.0
	if (a <= -21000000.0)
		tmp = t_0;
	elseif (a <= 4.8e+33)
		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 <= -21000000.0)
		tmp = t_0;
	elseif (a <= 4.8e+33)
		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[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -21000000.0], t$95$0, If[LessEqual[a, 4.8e+33], 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 < -2.1e7 or 4.8e33 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 44.9

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

   4. Applied rewrites 44.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up 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. pow2 N/A

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

      6. associate-*r* N/A

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

      7. pow3 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 44.9

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

   6. Applied rewrites 44.9%

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

   if -2.1e7 < a < 4.8e33

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 70.0

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

   6. Applied rewrites 70.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \left(b \cdot b\right) \cdot 4 - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 51.8%

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

Alternative 8: 81.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -21000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+33}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -21000000.0)
     t_0
     (if (<= a 4.8e+33) (- (* (* 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 <= -21000000.0) {
		tmp = t_0;
	} else if (a <= 4.8e+33) {
		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 <= (-21000000.0d0)) then
        tmp = t_0
    else if (a <= 4.8d+33) 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 <= -21000000.0) {
		tmp = t_0;
	} else if (a <= 4.8e+33) {
		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 <= -21000000.0:
		tmp = t_0
	elif a <= 4.8e+33:
		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 <= -21000000.0)
		tmp = t_0;
	elseif (a <= 4.8e+33)
		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 <= -21000000.0)
		tmp = t_0;
	elseif (a <= 4.8e+33)
		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, -21000000.0], t$95$0, If[LessEqual[a, 4.8e+33], 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 < -2.1e7 or 4.8e33 < a

   1. Initial program 74.4%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 44.9

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

   4. Applied rewrites 44.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up 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. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 44.9

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

   6. Applied rewrites 44.9%

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

   if -2.1e7 < a < 4.8e33

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 70.0

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

   6. Applied rewrites 70.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \left(b \cdot b\right) \cdot 4 - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 51.8%

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

Alternative 9: 61.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.5e+153) (- (* (* a a) 4.0) 1.0) (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.5e+153) {
		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 <= 6.5d+153) 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 <= 6.5e+153) {
		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 <= 6.5e+153:
		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 <= 6.5e+153)
		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 <= 6.5e+153)
		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, 6.5e+153], 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 < 6.49999999999999972e153

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-pow.f64 53.2

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

   4. Applied rewrites 53.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-lft1-in N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 53.2

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

      16. lift-pow.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow-prod-up N/A

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

      19. pow2 N/A

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

      20. pow2 N/A

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

      21. associate-*r* N/A

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

      22. pow3 N/A

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

      23. lower-*.f64 N/A

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

      24. pow3 N/A

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

      25. lower-*.f64 N/A

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

      26. lift-*.f64 53.1

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

   6. Applied rewrites 53.1%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 51.7

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

   9. Applied rewrites 51.7%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 51.7

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

   11. Applied rewrites 51.7%

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

   if 6.49999999999999972e153 < b

   1. Initial program 74.4%

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

   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. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 70.0

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

   6. Applied rewrites 70.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \left(b \cdot b\right) \cdot 4 - 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 51.8%

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

Alternative 10: 51.7% 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 74.4%

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

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. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-pow.f64 53.2

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

4. Applied rewrites 53.2%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow2 N/A

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

   6. lift-+.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-lft1-in N/A

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

   14. lift-fma.f64 N/A

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

   15. lift-*.f64 53.2

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

   16. lift-pow.f64 N/A

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

   17. metadata-eval N/A

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

   18. pow-prod-up N/A

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

   19. pow2 N/A

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

   20. pow2 N/A

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

   21. associate-*r* N/A

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

   22. pow3 N/A

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

   23. lower-*.f64 N/A

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

   24. pow3 N/A

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

   25. lower-*.f64 N/A

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

   26. lift-*.f64 53.1

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

6. Applied rewrites 53.1%

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

7. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 51.7

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

9. Applied rewrites 51.7%

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

    1. lift-*.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. pow2 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lift-*.f64 51.7

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

11. Applied rewrites 51.7%

    \[\leadsto \left(a \cdot a\right) \cdot 4 - 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025152 
(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))