Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.6% → 98.3%

Time: 4.2s

Alternatives: 13

Speedup: 2.1×

• 39.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\\ \mathbf{if}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot \left(a - 4\right)\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0
         (-
          (+
           (pow (+ (* a a) (* b b)) 2.0)
           (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
          1.0)))
   (if (<= t_0 INFINITY) t_0 (- (* a (* a (+ 4.0 (* a (- a 4.0))))) 1.0))))

C

double code(double a, double b) {
	double t_0 = (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (a * (a * (4.0 + (a * (a - 4.0))))) - 1.0;
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double t_0 = (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = (a * (a * (4.0 + (a * (a - 4.0))))) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = (a * (a * (4.0 + (a * (a - 4.0))))) - 1.0
	return tmp

Julia

function code(a, b)
	t_0 = 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(3.0 + a))))) - 1.0)
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a - 4.0))))) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0;
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = (a * (a * (4.0 + (a * (a - 4.0))))) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = 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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(a * N[(a * N[(4.0 + N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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 3 binary64) a))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 73.6%

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

   if +inf.0 < (-.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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.9%

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

   6. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 69.5%

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 69.5%

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

   11. Applied rewrites 69.5%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(\mathsf{fma}\left(\left(-3 - a\right) \cdot b, b, \left(a \cdot \left(a - 1\right)\right) \cdot a\right), -4, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot \left(a - 4\right)\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a))))
   (if (<=
        (-
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
         1.0)
        INFINITY)
     (fma
      t_0
      t_0
      (fma (fma (* (- -3.0 a) b) b (* (* a (- a 1.0)) a)) -4.0 -1.0))
     (- (* a (* a (+ 4.0 (* a (- a 4.0))))) 1.0))))

C

double code(double a, double b) {
	double t_0 = fma(b, b, (a * a));
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (3.0 + a))))) - 1.0) <= ((double) INFINITY)) {
		tmp = fma(t_0, t_0, fma(fma(((-3.0 - a) * b), b, ((a * (a - 1.0)) * a)), -4.0, -1.0));
	} else {
		tmp = (a * (a * (4.0 + (a * (a - 4.0))))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (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(3.0 + a))))) - 1.0) <= Inf)
		tmp = fma(t_0, t_0, fma(fma(Float64(Float64(-3.0 - a) * b), b, Float64(Float64(a * Float64(a - 1.0)) * a)), -4.0, -1.0));
	else
		tmp = Float64(Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a - 4.0))))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], Infinity], N[(t$95$0 * t$95$0 + N[(N[(N[(N[(-3.0 - a), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * N[(a - 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -4.0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(a * N[(4.0 + N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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 3 binary64) a))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 73.6%

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

   3. Applied rewrites 75.0%

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

   if +inf.0 < (-.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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.9%

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

   6. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 69.5%

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 69.5%

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

   11. Applied rewrites 69.5%

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

Alternative 3: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2800000000:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot \left(a - 4\right)\right)\right) - 1\\ \mathbf{elif}\;a \leq 14.2:\\ \;\;\;\;\mathsf{fma}\left(12, \frac{1}{{b}^{-2}}, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(1 - a, 4, a \cdot a\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2800000000.0)
   (- (* a (* a (+ 4.0 (* a (- a 4.0))))) 1.0)
   (if (<= a 14.2)
     (- (fma 12.0 (/ 1.0 (pow b -2.0)) (pow b 4.0)) 1.0)
     (- (* a (* a (fma (- 1.0 a) 4.0 (* a a)))) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2800000000.0) {
		tmp = (a * (a * (4.0 + (a * (a - 4.0))))) - 1.0;
	} else if (a <= 14.2) {
		tmp = fma(12.0, (1.0 / pow(b, -2.0)), pow(b, 4.0)) - 1.0;
	} else {
		tmp = (a * (a * fma((1.0 - a), 4.0, (a * a)))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2800000000.0)
		tmp = Float64(Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a - 4.0))))) - 1.0);
	elseif (a <= 14.2)
		tmp = Float64(fma(12.0, Float64(1.0 / (b ^ -2.0)), (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(Float64(a * Float64(a * fma(Float64(1.0 - a), 4.0, Float64(a * a)))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2800000000.0], N[(N[(a * N[(a * N[(4.0 + N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 14.2], N[(N[(12.0 * N[(1.0 / N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * N[(a * N[(N[(1.0 - a), $MachinePrecision] * 4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.8e9

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.9%

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

   6. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 69.5%

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 69.5%

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

   11. Applied rewrites 69.5%

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

   if -2.8e9 < a < 14.199999999999999

   1. Initial program 73.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.2%

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

   4. Applied rewrites 70.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-pow.f64 N/A

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

      7. metadata-eval 70.2%

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

   6. Applied rewrites 70.2%

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

   if 14.199999999999999 < a

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.9%

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

   6. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 69.5%

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

   8. Applied rewrites 69.5%

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

Alternative 4: 94.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2800000000:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot \left(a - 4\right)\right)\right) - 1\\ \mathbf{elif}\;a \leq 14.2:\\ \;\;\;\;\mathsf{fma}\left(12, b \cdot b, {b}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(1 - a, 4, a \cdot a\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2800000000.0)
   (- (* a (* a (+ 4.0 (* a (- a 4.0))))) 1.0)
   (if (<= a 14.2)
     (- (fma 12.0 (* b b) (pow b 4.0)) 1.0)
     (- (* a (* a (fma (- 1.0 a) 4.0 (* a a)))) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2800000000.0) {
		tmp = (a * (a * (4.0 + (a * (a - 4.0))))) - 1.0;
	} else if (a <= 14.2) {
		tmp = fma(12.0, (b * b), pow(b, 4.0)) - 1.0;
	} else {
		tmp = (a * (a * fma((1.0 - a), 4.0, (a * a)))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2800000000.0)
		tmp = Float64(Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a - 4.0))))) - 1.0);
	elseif (a <= 14.2)
		tmp = Float64(fma(12.0, Float64(b * b), (b ^ 4.0)) - 1.0);
	else
		tmp = Float64(Float64(a * Float64(a * fma(Float64(1.0 - a), 4.0, Float64(a * a)))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2800000000.0], N[(N[(a * N[(a * N[(4.0 + N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 14.2], N[(N[(12.0 * N[(b * b), $MachinePrecision] + N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * N[(a * N[(N[(1.0 - a), $MachinePrecision] * 4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.8e9

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.9%

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

   6. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 69.5%

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 69.5%

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

   11. Applied rewrites 69.5%

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

   if -2.8e9 < a < 14.199999999999999

   1. Initial program 73.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.2%

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

   4. Applied rewrites 70.2%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 70.2%

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

   6. Applied rewrites 70.2%

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

   if 14.199999999999999 < a

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.9%

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

   6. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 69.5%

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

   8. Applied rewrites 69.5%

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

Alternative 5: 94.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2800000000:\\ \;\;\;\;a \cdot \left(a \cdot \left(4 + a \cdot \left(a - 4\right)\right)\right) - 1\\ \mathbf{elif}\;a \leq 14.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, 12\right)}{\frac{1}{b \cdot b}} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \mathsf{fma}\left(1 - a, 4, a \cdot a\right)\right) - 1\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2800000000.0)
   (- (* a (* a (+ 4.0 (* a (- a 4.0))))) 1.0)
   (if (<= a 14.2)
     (- (/ (fma b b 12.0) (/ 1.0 (* b b))) 1.0)
     (- (* a (* a (fma (- 1.0 a) 4.0 (* a a)))) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2800000000.0) {
		tmp = (a * (a * (4.0 + (a * (a - 4.0))))) - 1.0;
	} else if (a <= 14.2) {
		tmp = (fma(b, b, 12.0) / (1.0 / (b * b))) - 1.0;
	} else {
		tmp = (a * (a * fma((1.0 - a), 4.0, (a * a)))) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2800000000.0)
		tmp = Float64(Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a - 4.0))))) - 1.0);
	elseif (a <= 14.2)
		tmp = Float64(Float64(fma(b, b, 12.0) / Float64(1.0 / Float64(b * b))) - 1.0);
	else
		tmp = Float64(Float64(a * Float64(a * fma(Float64(1.0 - a), 4.0, Float64(a * a)))) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2800000000.0], N[(N[(a * N[(a * N[(4.0 + N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 14.2], N[(N[(N[(b * b + 12.0), $MachinePrecision] / N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(a * N[(a * N[(N[(1.0 - a), $MachinePrecision] * 4.0 + N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.8e9

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.9%

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

   6. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 69.5%

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 69.5%

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

   11. Applied rewrites 69.5%

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

   if -2.8e9 < a < 14.199999999999999

   1. Initial program 73.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.2%

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

   4. Applied rewrites 70.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-pow.f64 N/A

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

      7. metadata-eval 70.2%

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

   6. Applied rewrites 70.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. add-to-fraction N/A

         \[\leadsto \frac{{b}^{4} \cdot {b}^{-2} + 12}{\color{blue}{{b}^{-2}}} - 1 \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{{b}^{4} \cdot {b}^{-2} + 12}{\color{blue}{{b}^{-2}}} - 1 \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{{b}^{4} \cdot {b}^{-2} + 12}{{b}^{-2}} - 1 \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{{b}^{4} \cdot {b}^{-2} + 12}{{b}^{-2}} - 1 \]

      9. pow-prod-up N/A

         \[\leadsto \frac{{b}^{\left(4 + -2\right)} + 12}{{b}^{-2}} - 1 \]

      10. metadata-eval N/A

          \[\leadsto \frac{{b}^{2} + 12}{{b}^{-2}} - 1 \]

      11. metadata-eval N/A

          \[\leadsto \frac{{b}^{\left(\mathsf{neg}\left(-2\right)\right)} + 12}{{b}^{-2}} - 1 \]

      12. metadata-eval N/A

          \[\leadsto \frac{{b}^{2} + 12}{{b}^{-2}} - 1 \]

      13. pow2 N/A

          \[\leadsto \frac{b \cdot b + 12}{{b}^{-2}} - 1 \]

      14. lower-fma.f64 70.1%

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

      15. lift-pow.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(b, b, 12\right)}{{b}^{\left(\mathsf{neg}\left(2\right)\right)}} - 1 \]

      17. pow-neg N/A

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

      18. lower-unsound-pow.f32 N/A

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

      19. lower-pow.f32 N/A

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

      20. pow2 N/A

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

      21. lower-unsound-/.f64 N/A

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

      22. lower-*.f64 70.1%

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

   8. Applied rewrites 70.1%

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

   if 14.199999999999999 < a

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.9%

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

   6. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 69.5%

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

   8. Applied rewrites 69.5%

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

Alternative 6: 94.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(4 + a \cdot \left(a - 4\right)\right)\right) - 1\\ \mathbf{if}\;a \leq -2800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 14.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, 12\right)}{\frac{1}{b \cdot b}} - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (* a (* a (+ 4.0 (* a (- a 4.0))))) 1.0)))
   (if (<= a -2800000000.0)
     t_0
     (if (<= a 14.2) (- (/ (fma b b 12.0) (/ 1.0 (* b b))) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * (a * (4.0 + (a * (a - 4.0))))) - 1.0;
	double tmp;
	if (a <= -2800000000.0) {
		tmp = t_0;
	} else if (a <= 14.2) {
		tmp = (fma(b, b, 12.0) / (1.0 / (b * b))) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * Float64(a * Float64(4.0 + Float64(a * Float64(a - 4.0))))) - 1.0)
	tmp = 0.0
	if (a <= -2800000000.0)
		tmp = t_0;
	elseif (a <= 14.2)
		tmp = Float64(Float64(fma(b, b, 12.0) / Float64(1.0 / Float64(b * b))) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * N[(a * N[(4.0 + N[(a * N[(a - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[a, -2800000000.0], t$95$0, If[LessEqual[a, 14.2], N[(N[(N[(b * b + 12.0), $MachinePrecision] / N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.8e9 or 14.199999999999999 < a

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 52.9%

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

   6. Applied rewrites 52.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 69.5%

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 69.5%

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

   11. Applied rewrites 69.5%

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

   if -2.8e9 < a < 14.199999999999999

   1. Initial program 73.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 70.2%

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

   4. Applied rewrites 70.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-pow.f64 N/A

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

      7. metadata-eval 70.2%

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

   6. Applied rewrites 70.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. add-to-fraction N/A

         \[\leadsto \frac{{b}^{4} \cdot {b}^{-2} + 12}{\color{blue}{{b}^{-2}}} - 1 \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{{b}^{4} \cdot {b}^{-2} + 12}{\color{blue}{{b}^{-2}}} - 1 \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{{b}^{4} \cdot {b}^{-2} + 12}{{b}^{-2}} - 1 \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{{b}^{4} \cdot {b}^{-2} + 12}{{b}^{-2}} - 1 \]

      9. pow-prod-up N/A

         \[\leadsto \frac{{b}^{\left(4 + -2\right)} + 12}{{b}^{-2}} - 1 \]

      10. metadata-eval N/A

          \[\leadsto \frac{{b}^{2} + 12}{{b}^{-2}} - 1 \]

      11. metadata-eval N/A

          \[\leadsto \frac{{b}^{\left(\mathsf{neg}\left(-2\right)\right)} + 12}{{b}^{-2}} - 1 \]

      12. metadata-eval N/A

          \[\leadsto \frac{{b}^{2} + 12}{{b}^{-2}} - 1 \]

      13. pow2 N/A

          \[\leadsto \frac{b \cdot b + 12}{{b}^{-2}} - 1 \]

      14. lower-fma.f64 70.1%

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

      15. lift-pow.f64 N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(b, b, 12\right)}{{b}^{\left(\mathsf{neg}\left(2\right)\right)}} - 1 \]

      17. pow-neg N/A

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

      18. lower-unsound-pow.f32 N/A

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

      19. lower-pow.f32 N/A

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

      20. pow2 N/A

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

      21. lower-unsound-/.f64 N/A

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

      22. lower-*.f64 70.1%

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

   8. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b, 12\right)}{\frac{1}{b \cdot b}}} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

2. Taylor expanded in b around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-pow.f64 52.9%

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

4. Applied rewrites 52.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. pow-prod-down N/A

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

   7. lift-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 52.9%

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

6. Applied rewrites 52.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-rgt-out N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift--.f64 69.5%

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

8. Applied rewrites 69.5%

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

9. Taylor expanded in a around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 69.5%

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

11. Applied rewrites 69.5%

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

Alternative 8: 69.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $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 3 binary64) a))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 51.7%

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

   7. Applied rewrites 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.7%

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

   9. Applied rewrites 51.7%

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

   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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 73.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.3%

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

   4. Applied rewrites 45.3%

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

Alternative 9: 69.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[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[(3.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $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 3 binary64) a))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 51.7%

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

   7. Applied rewrites 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.7%

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

   9. Applied rewrites 51.7%

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

   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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 73.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.3%

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

   4. Applied rewrites 45.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.3%

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

   6. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-prod-down N/A

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

      5. pow-sqr N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 45.3%

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

      11. lift-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow-neg N/A

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{{a}^{4}}}} \]

      14. lower-unsound-pow.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{{a}^{\color{blue}{4}}}} \]

      15. lower-pow.f32 N/A

          \[\leadsto \frac{1}{\frac{1}{{a}^{\color{blue}{4}}}} \]

      16. metadata-eval N/A

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

      17. pow-add N/A

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

      18. unpow-prod-down N/A

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

      19. lift-*.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 N/A

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

      22. lower-unsound-/.f64 45.3%

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. associate-*l* N/A

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

      26. *-commutative N/A

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

      27. lower-*.f64 N/A

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

      28. *-commutative N/A

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

      29. lower-*.f64 45.3%

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

   8. Applied rewrites 45.3%

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

Alternative 10: 69.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (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

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

Python

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

Julia

function code(a, b)
	tmp = 0.0
	if (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(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

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (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

code[a_, b_] := If[LessEqual[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[(3.0 + a), $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 3 binary64) a))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 51.7%

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

   7. Applied rewrites 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.7%

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

   9. Applied rewrites 51.7%

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

   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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 73.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.3%

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

   4. Applied rewrites 45.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.3%

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

   6. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 45.3%

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

   8. Applied rewrites 45.3%

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

Alternative 11: 69.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((((((a * a) + (b * b)) ** 2.0d0) + (4.0d0 * (((a * a) * (1.0d0 - a)) + ((b * b) * (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

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

Python

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

Julia

function code(a, b)
	tmp = 0.0
	if (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(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

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (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

code[a_, b_] := If[LessEqual[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[(3.0 + a), $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 3 binary64) a))))) #s(literal 1 binary64)) < -0.5

   1. Initial program 73.6%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-pow.f64 52.9%

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 51.7%

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

   7. Applied rewrites 51.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.7%

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

   9. Applied rewrites 51.7%

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

   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 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 73.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-pow.f64 45.3%

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

   4. Applied rewrites 45.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.3%

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

   6. Applied rewrites 45.3%

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

Alternative 12: 68.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

2. Taylor expanded in b around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-pow.f64 52.9%

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

4. Applied rewrites 52.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. pow-prod-down N/A

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

   7. lift-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 52.9%

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

6. Applied rewrites 52.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-rgt-out N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lift--.f64 69.5%

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

8. Applied rewrites 69.5%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 68.7%

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

Alternative 13: 51.7% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.6%

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

2. Taylor expanded in b around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-pow.f64 52.9%

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

4. Applied rewrites 52.9%

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

5. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 51.7%

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

7. Applied rewrites 51.7%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 51.7%

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

9. Applied rewrites 51.7%

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

Reproduce

herbie shell --seed 2025185 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))