Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.8% → 99.2%

Time: 11.2s

Alternatives: 12

Speedup: 4.1×

• 61.0% 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: 73.8% 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: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 9.7 \cdot 10^{-11}:\\ \;\;\;\;\left(\mathsf{fma}\left(a - -4, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a + a, a, b\_m \cdot b\_m\right) \cdot b\_m, b\_m, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 9.7e-11)
   (- (* (* (fma (- a -4.0) a 4.0) a) a) 1.0)
   (fma (* (fma (+ a a) a (* b_m b_m)) b_m) b_m (* (* (* a a) a) a))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 9.7e-11) {
		tmp = ((fma((a - -4.0), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = fma((fma((a + a), a, (b_m * b_m)) * b_m), b_m, (((a * a) * a) * a));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 9.7e-11)
		tmp = Float64(Float64(Float64(fma(Float64(a - -4.0), a, 4.0) * a) * a) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(a + a), a, Float64(b_m * b_m)) * b_m), b_m, Float64(Float64(Float64(a * a) * a) * a));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 9.7e-11], N[(N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a + a), $MachinePrecision] * a + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m + N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.7000000000000001e-11

   1. Initial program 83.6%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

   if 9.7000000000000001e-11 < b

   1. Initial program 64.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in b around inf

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. pow3 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 97.5

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

   9. Applied rewrites 97.5%

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

Alternative 2: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \mathsf{fma}\left(\mathsf{fma}\left(a + a, a, b\_m \cdot b\_m\right) \cdot b\_m, b\_m, \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, a, 4\right)\right) - 1\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (fma
  (* (fma (+ a a) a (* b_m b_m)) b_m)
  b_m
  (- (* (* a a) (fma a a (fma 4.0 a 4.0))) 1.0)))

C

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

Julia

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := N[(N[(N[(N[(a + a), $MachinePrecision] * a + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m + N[(N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(4.0 * a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in b around inf

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

   1. pow2 N/A

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

   2. +-commutative N/A

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

   3. pow2 N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 99.2

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

6. Applied rewrites 99.2%

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

Alternative 3: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \mathsf{fma}\left(\left(b\_m \cdot b\_m\right) \cdot b\_m, b\_m, \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(4, a, 4\right)\right) - 1\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (fma (* (* b_m b_m) b_m) b_m (- (* (* a a) (fma a a (fma 4.0 a 4.0))) 1.0)))

C

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

Julia

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m + N[(N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(4.0 * a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in b around inf

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

   1. pow2 N/A

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

   2. lift-*.f64 99.1

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

6. Applied rewrites 99.1%

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

Alternative 4: 98.7% accurate, 1.9× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (let* ((t_0 (fma b_m b_m (* a a))))
   (fma t_0 t_0 (- (* (* b_m b_m) 4.0) 1.0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

2. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 99.0

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

4. Applied rewrites 99.0%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate--l+ N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

6. Applied rewrites 99.0%

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

Alternative 5: 97.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;\left(\mathsf{fma}\left(a - -4, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a + a, a, b\_m \cdot b\_m\right) \cdot b\_m, b\_m, -1\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 3.8e+23)
   (- (* (* (fma (- a -4.0) a 4.0) a) a) 1.0)
   (fma (* (fma (+ a a) a (* b_m b_m)) b_m) b_m -1.0)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 3.8e+23) {
		tmp = ((fma((a - -4.0), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = fma((fma((a + a), a, (b_m * b_m)) * b_m), b_m, -1.0);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 3.8e+23)
		tmp = Float64(Float64(Float64(fma(Float64(a - -4.0), a, 4.0) * a) * a) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(a + a), a, Float64(b_m * b_m)) * b_m), b_m, -1.0);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 3.8e+23], N[(N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(a + a), $MachinePrecision] * a + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.79999999999999975e23

   1. Initial program 83.4%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 95.9%

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

   if 3.79999999999999975e23 < b

   1. Initial program 63.0%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in b around inf

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 98.3%

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

Alternative 6: 94.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;\left(\mathsf{fma}\left(a - -4, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot b\_m\right) \cdot b\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 2.5e+30)
   (- (* (* (fma (- a -4.0) a 4.0) a) a) 1.0)
   (* (* (* b_m b_m) b_m) b_m)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 2.5e+30) {
		tmp = ((fma((a - -4.0), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((b_m * b_m) * b_m) * b_m;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 2.5e+30)
		tmp = Float64(Float64(Float64(fma(Float64(a - -4.0), a, 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(b_m * b_m) * b_m) * b_m);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 2.5e+30], N[(N[(N[(N[(N[(a - -4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4999999999999999e30

   1. Initial program 83.3%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 94.7

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

   4. Applied rewrites 94.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 94.7%

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

   if 2.4999999999999999e30 < b

   1. Initial program 62.5%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. cube-unmult N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 93.3

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

   4. Applied rewrites 93.3%

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

Alternative 7: 93.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq -900000:\\ \;\;\;\;\left(a - -4\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+39}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot b\_m\right) \cdot b\_m - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= a -900000.0)
   (* (- a -4.0) (* (* a a) a))
   (if (<= a 1.3e+39) (- (* (* (* b_m b_m) b_m) b_m) 1.0) (pow a 4.0))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (a <= -900000.0) {
		tmp = (a - -4.0) * ((a * a) * a);
	} else if (a <= 1.3e+39) {
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Fortran

b_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if (a <= (-900000.0d0)) then
        tmp = (a - (-4.0d0)) * ((a * a) * a)
    else if (a <= 1.3d+39) then
        tmp = (((b_m * b_m) * b_m) * b_m) - 1.0d0
    else
        tmp = a ** 4.0d0
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (a <= -900000.0) {
		tmp = (a - -4.0) * ((a * a) * a);
	} else if (a <= 1.3e+39) {
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0;
	} else {
		tmp = Math.pow(a, 4.0);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if a <= -900000.0:
		tmp = (a - -4.0) * ((a * a) * a)
	elif a <= 1.3e+39:
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0
	else:
		tmp = math.pow(a, 4.0)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (a <= -900000.0)
		tmp = Float64(Float64(a - -4.0) * Float64(Float64(a * a) * a));
	elseif (a <= 1.3e+39)
		tmp = Float64(Float64(Float64(Float64(b_m * b_m) * b_m) * b_m) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if (a <= -900000.0)
		tmp = (a - -4.0) * ((a * a) * a);
	elseif (a <= 1.3e+39)
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0;
	else
		tmp = a ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[a, -900000.0], N[(N[(a - -4.0), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.3e+39], N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9e5

   1. Initial program 32.5%

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

   2. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot {a}^{2} + 4 \cdot {a}^{3} \]

      9. cube-mult N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. fp-cancel-sign-sub N/A

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

      15. pow2 N/A

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

      16. sqr-neg-rev N/A

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

      17. lower-*.f64 N/A

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

   4. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. unpow3 N/A

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

      14. pow3 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lift--.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 90.9

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

   6. Applied rewrites 90.9%

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

   if -9e5 < a < 1.3e39

   1. Initial program 97.9%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. cube-unmult N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 1.3e39 < a

   1. Initial program 58.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 94.5

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

   6. Applied rewrites 94.5%

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

Alternative 8: 93.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot a\\ \mathbf{if}\;a \leq -900000:\\ \;\;\;\;\left(a - -4\right) \cdot t\_0\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+39}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot b\_m\right) \cdot b\_m - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot a\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (let* ((t_0 (* (* a a) a)))
   (if (<= a -900000.0)
     (* (- a -4.0) t_0)
     (if (<= a 1.3e+39) (- (* (* (* b_m b_m) b_m) b_m) 1.0) (* t_0 a)))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double t_0 = (a * a) * a;
	double tmp;
	if (a <= -900000.0) {
		tmp = (a - -4.0) * t_0;
	} else if (a <= 1.3e+39) {
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0;
	} else {
		tmp = t_0 * a;
	}
	return tmp;
}

Fortran

b_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * a
    if (a <= (-900000.0d0)) then
        tmp = (a - (-4.0d0)) * t_0
    else if (a <= 1.3d+39) then
        tmp = (((b_m * b_m) * b_m) * b_m) - 1.0d0
    else
        tmp = t_0 * a
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double t_0 = (a * a) * a;
	double tmp;
	if (a <= -900000.0) {
		tmp = (a - -4.0) * t_0;
	} else if (a <= 1.3e+39) {
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0;
	} else {
		tmp = t_0 * a;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	t_0 = (a * a) * a
	tmp = 0
	if a <= -900000.0:
		tmp = (a - -4.0) * t_0
	elif a <= 1.3e+39:
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0
	else:
		tmp = t_0 * a
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	t_0 = Float64(Float64(a * a) * a)
	tmp = 0.0
	if (a <= -900000.0)
		tmp = Float64(Float64(a - -4.0) * t_0);
	elseif (a <= 1.3e+39)
		tmp = Float64(Float64(Float64(Float64(b_m * b_m) * b_m) * b_m) - 1.0);
	else
		tmp = Float64(t_0 * a);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	t_0 = (a * a) * a;
	tmp = 0.0;
	if (a <= -900000.0)
		tmp = (a - -4.0) * t_0;
	elseif (a <= 1.3e+39)
		tmp = (((b_m * b_m) * b_m) * b_m) - 1.0;
	else
		tmp = t_0 * a;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -900000.0], N[(N[(a - -4.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[a, 1.3e+39], N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] - 1.0), $MachinePrecision], N[(t$95$0 * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9e5

   1. Initial program 32.5%

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

   2. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot {a}^{2} + 4 \cdot {a}^{3} \]

      9. cube-mult N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. fp-cancel-sign-sub N/A

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

      15. pow2 N/A

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

      16. sqr-neg-rev N/A

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

      17. lower-*.f64 N/A

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

   4. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. unpow3 N/A

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

      14. pow3 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. metadata-eval N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lift--.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 90.9

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

   6. Applied rewrites 90.9%

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

   if -9e5 < a < 1.3e39

   1. Initial program 97.9%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. cube-unmult N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if 1.3e39 < a

   1. Initial program 58.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 94.5

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

   4. Applied rewrites 94.5%

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

Alternative 9: 81.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.15 \cdot 10^{+25}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b\_m \cdot b\_m\right) \cdot b\_m\right) \cdot b\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<= b_m 1.15e+25) (- (* (* a a) 4.0) 1.0) (* (* (* b_m b_m) b_m) b_m)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.15e+25) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b_m * b_m) * b_m) * b_m;
	}
	return tmp;
}

Fortran

b_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if (b_m <= 1.15d+25) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((b_m * b_m) * b_m) * b_m
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (b_m <= 1.15e+25) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b_m * b_m) * b_m) * b_m;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if b_m <= 1.15e+25:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b_m * b_m) * b_m) * b_m
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (b_m <= 1.15e+25)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b_m * b_m) * b_m) * b_m);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if (b_m <= 1.15e+25)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b_m * b_m) * b_m) * b_m;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[b$95$m, 1.15e+25], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999e25

   1. Initial program 83.3%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 72.2%

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

   if 1.1499999999999999e25 < b

   1. Initial program 62.9%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. cube-unmult N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. cube-unmult N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 92.6

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

   4. Applied rewrites 92.6%

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

Alternative 10: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b\_m \cdot b\_m\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b\_m \cdot b\_m\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq -0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b_m b_m)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b_m b_m) (- 1.0 (* 3.0 a))))))
       1.0)
      -0.5)
   (- (* (* a a) 4.0) 1.0)
   (* (* (* a a) a) a)))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (((pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Fortran

b_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if ((((((a * a) + (b_m * b_m)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b_m * b_m) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= (-0.5d0)) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((a * a) * a) * a
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (((Math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if ((math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((a * a) * a) * a
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b_m * b_m)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b_m * b_m) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= -0.5)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if ((((((a * a) + (b_m * b_m)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((a * a) * a) * a;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 98.4%

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

   if -0.5 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 65.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 58.6

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

   4. Applied rewrites 58.6%

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

Alternative 11: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;\left({\left(a \cdot a + b\_m \cdot b\_m\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b\_m \cdot b\_m\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \leq -0.5:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m)
 :precision binary64
 (if (<=
      (-
       (+
        (pow (+ (* a a) (* b_m b_m)) 2.0)
        (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b_m b_m) (- 1.0 (* 3.0 a))))))
       1.0)
      -0.5)
   (- (* (* a a) 4.0) 1.0)
   (* (* a a) (* a a))))

C

b_m = fabs(b);
double code(double a, double b_m) {
	double tmp;
	if (((pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Fortran

b_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if ((((((a * a) + (b_m * b_m)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 + a)) + ((b_m * b_m) * (1.0d0 - (3.0d0 * a)))))) - 1.0d0) <= (-0.5d0)) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (a * a) * (a * a)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	double tmp;
	if (((Math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	tmp = 0
	if ((math.pow(((a * a) + (b_m * b_m)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (a * a) * (a * a)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m)
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b_m * b_m)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b_m * b_m) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0) <= -0.5)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m)
	tmp = 0.0;
	if ((((((a * a) + (b_m * b_m)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b_m * b_m) * (1.0 - (3.0 * a)))))) - 1.0) <= -0.5)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (a * a) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 98.4%

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

   if -0.5 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (+.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (-.f64 #s(literal 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 65.0%

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

   2. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. associate-*l* N/A

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

      6. inv-pow N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot {a}^{2} + 4 \cdot {a}^{3} \]

      9. cube-mult N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out N/A

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

      13. *-commutative N/A

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

      14. fp-cancel-sign-sub N/A

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

      15. pow2 N/A

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

      16. sqr-neg-rev N/A

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

      17. lower-*.f64 N/A

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

   4. Applied rewrites 59.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 58.6%

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

Alternative 12: 50.7% accurate, 5.9× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m) :precision binary64 (- (* (* a a) 4.0) 1.0))

C

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

Fortran

b_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    code = ((a * a) * 4.0d0) - 1.0d0
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m) {
	return ((a * a) * 4.0) - 1.0;
}

Python

b_m = math.fabs(b)
def code(a, b_m):
	return ((a * a) * 4.0) - 1.0

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_] := N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 73.8%

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

2. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt1-in N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. lower-fma.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 69.1

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

4. Applied rewrites 69.1%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 50.7%

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

Reproduce

herbie shell --seed 2025130 
(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))