Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.4% → 99.1%

Time: 3.0s

Alternatives: 11

Speedup: 6.4×

• 61.4% 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 74.5%

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

4. Applied rewrites 75.6%

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

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 99.1

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

7. Applied rewrites 99.1%

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

Alternative 2: 97.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-51}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \left(b \cdot b\right) \cdot 2\right) + 4\right) \cdot \left(a \cdot a\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.25e-5)
   (- (* (* (fma (+ 4.0 a) a (fma (* b b) 2.0 4.0)) a) a) 1.0)
   (if (<= a 8.2e-51)
     (- (* (* (fma b b 4.0) b) b) 1.0)
     (- (* (+ (fma (+ 4.0 a) a (* (* b b) 2.0)) 4.0) (* a a)) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.25e-5) {
		tmp = ((fma((4.0 + a), a, fma((b * b), 2.0, 4.0)) * a) * a) - 1.0;
	} else if (a <= 8.2e-51) {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	} else {
		tmp = ((fma((4.0 + a), a, ((b * b) * 2.0)) + 4.0) * (a * a)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.25e-5)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0)) * a) * a) - 1.0);
	elseif (a <= 8.2e-51)
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, Float64(Float64(b * b) * 2.0)) + 4.0) * Float64(a * a)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.25e-5], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 8.2e-51], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.25000000000000014e-5

   1. Initial program 40.4%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 94.6

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

   8. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 94.7%

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

   if -2.25000000000000014e-5 < a < 8.19999999999999947e-51

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if 8.19999999999999947e-51 < a

   1. Initial program 63.0%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 3: 97.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-5} \lor \neg \left(a \leq 8.2 \cdot 10^{-51}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.25e-5) (not (<= a 8.2e-51)))
   (- (* (* (fma (+ 4.0 a) a (fma (* b b) 2.0 4.0)) a) a) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.25e-5) || !(a <= 8.2e-51)) {
		tmp = ((fma((4.0 + a), a, fma((b * b), 2.0, 4.0)) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.25e-5) || !(a <= 8.2e-51))
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0)) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.25e-5], N[Not[LessEqual[a, 8.2e-51]], $MachinePrecision]], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.25000000000000014e-5 or 8.19999999999999947e-51 < a

   1. Initial program 51.4%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 97.2

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

   8. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 97.2%

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

   if -2.25000000000000014e-5 < a < 8.19999999999999947e-51

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 99.6

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

   9. Applied rewrites 99.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.25 \cdot 10^{-5} \lor \neg \left(a \leq 8.2 \cdot 10^{-51}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.4 \cdot 10^{+17}:\\ \;\;\;\;\left(\mathsf{fma}\left(4 + a, a, \mathsf{fma}\left(b \cdot b, 2, 4\right)\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, a \cdot a\right), b \cdot b, \left(b \cdot b\right) \cdot 4 - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.4e+17)
   (- (* (* (fma (+ 4.0 a) a (fma (* b b) 2.0 4.0)) a) a) 1.0)
   (fma (fma b b (* a a)) (* b b) (- (* (* b b) 4.0) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.4e+17) {
		tmp = ((fma((4.0 + a), a, fma((b * b), 2.0, 4.0)) * a) * a) - 1.0;
	} 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 (b <= 7.4e+17)
		tmp = Float64(Float64(Float64(fma(Float64(4.0 + a), a, fma(Float64(b * b), 2.0, 4.0)) * a) * a) - 1.0);
	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[LessEqual[b, 7.4e+17], N[(N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $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 b < 7.4e17

   1. Initial program 78.3%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   10. Applied rewrites 85.8%

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

   if 7.4e17 < b

   1. Initial program 64.2%

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

   4. Applied rewrites 68.4%

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

   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 99.9

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

   7. Applied rewrites 99.9%

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

   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 99.3

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

   10. Applied rewrites 99.3%

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

Alternative 5: 81.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(4 + a, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.2e+18)
   (- (* (fma (+ 4.0 a) a 4.0) (* a a)) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.2e+18) {
		tmp = (fma((4.0 + a), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.2e+18)
		tmp = Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.2e+18], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2e18

   1. Initial program 78.3%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift-+.f64 79.9

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

   11. Applied rewrites 79.9%

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

   if 5.2e18 < b

   1. Initial program 64.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 96.6

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

   7. Applied rewrites 96.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 96.6

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

   9. Applied rewrites 96.6%

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

Alternative 6: 81.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+34} \lor \neg \left(a \leq 360000000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -2.7e+34) (not (<= a 360000000.0)))
   (* (* a a) (* a a))
   (- (* (* b b) 4.0) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -2.7e+34) || !(a <= 360000000.0)) {
		tmp = (a * a) * (a * a);
	} 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 ((a <= (-2.7d+34)) .or. (.not. (a <= 360000000.0d0))) then
        tmp = (a * a) * (a * a)
    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 ((a <= -2.7e+34) || !(a <= 360000000.0)) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * 4.0) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a <= -2.7e+34) or not (a <= 360000000.0):
		tmp = (a * a) * (a * a)
	else:
		tmp = ((b * b) * 4.0) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -2.7e+34) || !(a <= 360000000.0))
		tmp = Float64(Float64(a * a) * Float64(a * a));
	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 ((a <= -2.7e+34) || ~((a <= 360000000.0)))
		tmp = (a * a) * (a * a);
	else
		tmp = ((b * b) * 4.0) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -2.7e+34], N[Not[LessEqual[a, 360000000.0]], $MachinePrecision]], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.7e34 or 3.6e8 < a

   1. Initial program 45.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 93.8

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

   5. Applied rewrites 93.8%

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 93.7

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

   7. Applied rewrites 93.7%

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

   if -2.7e34 < a < 3.6e8

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 69.4

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

   8. Applied rewrites 69.4%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+34} \lor \neg \left(a \leq 360000000\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot 4 - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.00025:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{+18}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.00025)
   (- (* (* a a) 4.0) 1.0)
   (if (<= b 8.2e+18) (* (* a a) (* a a)) (* (* b b) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.00025) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b <= 8.2e+18) {
		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 <= 0.00025d0) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else if (b <= 8.2d+18) 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 <= 0.00025) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b <= 8.2e+18) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.00025:
		tmp = ((a * a) * 4.0) - 1.0
	elif b <= 8.2e+18:
		tmp = (a * a) * (a * a)
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.00025)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	elseif (b <= 8.2e+18)
		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 <= 0.00025)
		tmp = ((a * a) * 4.0) - 1.0;
	elseif (b <= 8.2e+18)
		tmp = (a * a) * (a * a);
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.00025], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[b, 8.2e+18], 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 3 regimes

2. if b < 2.5000000000000001e-4

   1. Initial program 77.8%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 65.8

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

   8. Applied rewrites 65.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 59.2%

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

   if 2.5000000000000001e-4 < b < 8.2e18

   1. Initial program 99.2%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 61.2

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

   5. Applied rewrites 61.2%

      \[\leadsto \color{blue}{{a}^{4}} \]
   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 60.9

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

   7. Applied rewrites 60.9%

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

   if 8.2e18 < b

   1. Initial program 64.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{{b}^{4}} \]
   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 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 8: 80.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 4\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.2e+18)
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* (fma b b 4.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.2e+18) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((fma(b, b, 4.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.2e+18)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 4.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.2e+18], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2e18

   1. Initial program 78.3%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

   9. Taylor expanded in a around inf

      \[\leadsto {a}^{2} \cdot \left(a \cdot a\right) - 1 \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f64 78.9

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

   11. Applied rewrites 78.9%

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

   if 5.2e18 < b

   1. Initial program 64.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 96.6

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

   7. Applied rewrites 96.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-fma.f64 96.6

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

   9. Applied rewrites 96.6%

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

Alternative 9: 80.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+18}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.2e+18) (- (* (* a a) (* a a)) 1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.2e+18) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 5.2d+18) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5.2e+18) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5.2e+18:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.2e+18)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5.2e+18)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.2e+18], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.2e18

   1. Initial program 78.3%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

   9. Taylor expanded in a around inf

      \[\leadsto {a}^{2} \cdot \left(a \cdot a\right) - 1 \]
   10. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f64 78.9

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

   11. Applied rewrites 78.9%

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

   if 5.2e18 < b

   1. Initial program 64.2%

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

   3. Taylor expanded in b around inf

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

      1. lower-pow.f64 96.7

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

   5. Applied rewrites 96.7%

      \[\leadsto \color{blue}{{b}^{4}} \]
   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 96.6

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

   7. Applied rewrites 96.6%

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

Alternative 10: 60.3% accurate, 8.0× 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 77.3%

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

   3. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 53.1%

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

   if 6.49999999999999972e153 < b

   1. Initial program 60.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 11: 51.2% accurate, 11.4× 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 74.5%

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

3. Taylor expanded in a around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 63.7%

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

6. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. pow2 N/A

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

   6. +-commutative N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 64.8

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

8. Applied rewrites 64.8%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 49.4%

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

Reproduce

herbie shell --seed 2025083 
(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))