Bouland and Aaronson, Equation (25)

Percentage Accurate: 73.5% → 98.2%

Time: 3.8s

Alternatives: 12

Speedup: 4.1×

• 61.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double a, double b) {
	return (Math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}

Python

def code(a, b):
	return (math.pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0

Julia

function code(a, b)
	return Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 + a)) + Float64(Float64(b * b) * Float64(1.0 - Float64(3.0 * a)))))) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 + a), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 98.2% accurate, 0.5× speedup

Math

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

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) (- 1.0 (* 3.0 a))))))
          1.0)))
   (if (<= t_0 INFINITY) t_0 (- (* (fma (+ 4.0 a) a 4.0) (* a a)) 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) * (1.0 - (3.0 * a)))))) - 1.0;
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (fma((4.0 + a), a, 4.0) * (a * a)) - 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(1.0 - Float64(3.0 * a)))))) - 1.0)
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * Float64(a * a)) - 1.0);
	end
	return 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[(1.0 - N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $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 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 73.5%

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

   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 1 binary64) (*.f64 #s(literal 3 binary64) a)))))) #s(literal 1 binary64))

   1. Initial program 73.5%

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

   2. Taylor expanded in b around 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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 56.0

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 69.0

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

   7. Applied rewrites 69.0%

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

Alternative 2: 94.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(4 + a, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({b}^{3}, b, \left(b \cdot b\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.8e-10)
   (- (* (fma (+ 4.0 a) a 4.0) (* a a)) 1.0)
   (- (fma (pow b 3.0) b (* (* b b) 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.8e-10) {
		tmp = (fma((4.0 + a), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = fma(pow(b, 3.0), b, ((b * b) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.8e-10)
		tmp = Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(fma((b ^ 3.0), b, Float64(Float64(b * b) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.8e-10], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[Power[b, 3.0], $MachinePrecision] * b + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.7999999999999996e-10

   1. Initial program 73.5%

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

   2. Taylor expanded in b around 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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 56.0

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 69.0

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

   7. Applied rewrites 69.0%

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

   if 8.7999999999999996e-10 < b

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 69.4

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

   8. Applied rewrites 69.4%

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

Alternative 3: 94.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(4 + a, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot b\right) \cdot b, b, \left(b \cdot b\right) \cdot 4\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.8e-10)
   (- (* (fma (+ 4.0 a) a 4.0) (* a a)) 1.0)
   (- (fma (* (* b b) b) b (* (* b b) 4.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.8e-10) {
		tmp = (fma((4.0 + a), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = fma(((b * b) * b), b, ((b * b) * 4.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.8e-10)
		tmp = Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(fma(Float64(Float64(b * b) * b), b, Float64(Float64(b * b) * 4.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.8e-10], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.7999999999999996e-10

   1. Initial program 73.5%

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

   2. Taylor expanded in b around 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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 56.0

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 69.0

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

   7. Applied rewrites 69.0%

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

   if 8.7999999999999996e-10 < b

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

Alternative 4: 81.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(4 + a, a, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.8e-10)
   (- (* (fma (+ 4.0 a) a 4.0) (* a a)) 1.0)
   (fma (fma b b 4.0) (* b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.8e-10) {
		tmp = (fma((4.0 + a), a, 4.0) * (a * a)) - 1.0;
	} else {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.8e-10)
		tmp = Float64(Float64(fma(Float64(4.0 + a), a, 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.8e-10], N[(N[(N[(N[(4.0 + a), $MachinePrecision] * a + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.7999999999999996e-10

   1. Initial program 73.5%

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

   2. Taylor expanded in b around 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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 56.0

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 69.0

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

   7. Applied rewrites 69.0%

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

   if 8.7999999999999996e-10 < b

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

      1. sub-flip N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

Alternative 5: 80.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+23}:\\ \;\;\;\;{a}^{4}\\ \mathbf{elif}\;a \leq 7900:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 + a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e+23)
   (pow a 4.0)
   (if (<= a 7900.0)
     (fma (fma b b 4.0) (* b b) -1.0)
     (* (+ 4.0 a) (* (* a a) a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e+23) {
		tmp = pow(a, 4.0);
	} else if (a <= 7900.0) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = (4.0 + a) * ((a * a) * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.2e+23)
		tmp = a ^ 4.0;
	elseif (a <= 7900.0)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = Float64(Float64(4.0 + a) * Float64(Float64(a * a) * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.2e+23], N[Power[a, 4.0], $MachinePrecision], If[LessEqual[a, 7900.0], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 + a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.2e23

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   5. 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. unpow-prod-down N/A

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

      5. pow-prod-up N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 45.5

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

   6. Applied rewrites 45.5%

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

   if -1.2e23 < a < 7900

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

      1. sub-flip N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   if 7900 < a

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 45.8

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

   4. Applied rewrites 45.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 45.9

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

   7. Applied rewrites 45.9%

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

Alternative 6: 80.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.2 \cdot 10^{+23}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{elif}\;a \leq 7900:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, 4\right), b \cdot b, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4 + a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.2e+23)
   (* (* a a) (* a a))
   (if (<= a 7900.0)
     (fma (fma b b 4.0) (* b b) -1.0)
     (* (+ 4.0 a) (* (* a a) a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.2e+23) {
		tmp = (a * a) * (a * a);
	} else if (a <= 7900.0) {
		tmp = fma(fma(b, b, 4.0), (b * b), -1.0);
	} else {
		tmp = (4.0 + a) * ((a * a) * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.2e+23)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	elseif (a <= 7900.0)
		tmp = fma(fma(b, b, 4.0), Float64(b * b), -1.0);
	else
		tmp = Float64(Float64(4.0 + a) * Float64(Float64(a * a) * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.2e+23], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7900.0], N[(N[(b * b + 4.0), $MachinePrecision] * N[(b * b), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(4.0 + a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.2e23

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

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

   if -1.2e23 < a < 7900

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

      1. sub-flip N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   if 7900 < a

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 45.8

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

   4. Applied rewrites 45.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow3 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 45.9

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

   7. Applied rewrites 45.9%

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

Alternative 7: 80.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 114:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 114.0) (- (* (* a a) 4.0) 1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 114.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 114.0d0) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((b * b) * b) * b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 114.0) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 114.0:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * b) * b) * b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 114.0)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 114.0)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * b) * b) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 114.0], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 114

   1. Initial program 73.5%

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

   2. Taylor expanded in b around 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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 56.0

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 51.0

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

   7. Applied rewrites 51.0%

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

   if 114 < b

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 69.4

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

   6. Applied rewrites 69.4%

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

   7. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. pow3 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 46.1

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

   9. Applied rewrites 46.1%

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

Alternative 8: 72.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -310000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -310000000.0)
     t_0
     (if (<= a 1.6e+46) (fma (* b b) 4.0 -1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -310000000.0) {
		tmp = t_0;
	} else if (a <= 1.6e+46) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -310000000.0)
		tmp = t_0;
	elseif (a <= 1.6e+46)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -310000000.0], t$95$0, If[LessEqual[a, 1.6e+46], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.1e8 or 1.5999999999999999e46 < a

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 45.5

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

   4. Applied rewrites 45.5%

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

   if -3.1e8 < a < 1.5999999999999999e46

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

      1. sub-flip N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in b around 0

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

      1. sub-flip N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 50.4

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

   10. Applied rewrites 50.4%

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

Alternative 9: 66.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{+155}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot 4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.85e+155)
   (- (* (* a a) 4.0) 1.0)
   (if (<= a 6.2e+100) (fma (* b b) 4.0 -1.0) (* (* (* a a) a) 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.85e+155) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (a <= 6.2e+100) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = ((a * a) * a) * 4.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.85e+155)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	elseif (a <= 6.2e+100)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * 4.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.85e+155], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 6.2e+100], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * 4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8499999999999999e155

   1. Initial program 73.5%

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

   2. Taylor expanded in b around 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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 56.0

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 51.0

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

   7. Applied rewrites 51.0%

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

   if -1.8499999999999999e155 < a < 6.20000000000000014e100

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

      1. sub-flip N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in b around 0

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

      1. sub-flip N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 50.4

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

   10. Applied rewrites 50.4%

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

   if 6.20000000000000014e100 < a

   1. Initial program 73.5%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 45.8

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

   4. Applied rewrites 45.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 18.3

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

   7. Applied rewrites 18.3%

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

Alternative 10: 59.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+124}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.8e+124) (- (* (* a a) 4.0) 1.0) (fma (* b b) 4.0 -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.8e+124) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = fma((b * b), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.8e+124)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = fma(Float64(b * b), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.8e+124], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7999999999999998e124

   1. Initial program 73.5%

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

   2. Taylor expanded in b around 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. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 56.0

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 51.0

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

   7. Applied rewrites 51.0%

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

   if 3.7999999999999998e124 < b

   1. Initial program 73.5%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   4. Applied rewrites 69.4%

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

   5. Taylor expanded in a around 0

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

      1. sub-flip N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in b around 0

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

      1. sub-flip N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 50.4

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

   10. Applied rewrites 50.4%

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

Alternative 11: 50.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 69.4

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

4. Applied rewrites 69.4%

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

5. Taylor expanded in a around 0

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

   1. sub-flip N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. pow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 69.4

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

7. Applied rewrites 69.4%

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

8. Taylor expanded in b around 0

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

   1. sub-flip N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 50.4

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

10. Applied rewrites 50.4%

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

Alternative 12: 24.3% accurate, 56.6× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 -1.0)

C

double code(double a, double b) {
	return -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 = -1.0d0
end function

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 73.5%

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

2. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 69.4

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

4. Applied rewrites 69.4%

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

5. Taylor expanded in a around 0

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

   1. sub-flip N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. pow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 69.4

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

7. Applied rewrites 69.4%

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

8. Taylor expanded in b around 0

   \[\leadsto -1 \]
9. Step-by-step derivation

10. Applied rewrites 24.3%

    \[\leadsto -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))