Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 3.4s

Alternatives: 9

Speedup: 1.0×

• 60.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 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 * (b * b))) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = ((((a * a) + (b * b)) ^ 2.0) + (4.0 * (b * b))) - 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[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

Alternative 2: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 12.6:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(a, a + a, 4\right)}{b \cdot b} - -1\right) \cdot {b}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 12.6)
   (fma (* a a) (* a a) -1.0)
   (if (<= b 5e+153)
     (* (- (/ (fma a (+ a a) 4.0) (* b b)) -1.0) (pow b 4.0))
     (fma (* b b) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 12.6) {
		tmp = fma((a * a), (a * a), -1.0);
	} else if (b <= 5e+153) {
		tmp = ((fma(a, (a + a), 4.0) / (b * b)) - -1.0) * pow(b, 4.0);
	} else {
		tmp = fma((b * b), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 12.6)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	elseif (b <= 5e+153)
		tmp = Float64(Float64(Float64(fma(a, Float64(a + a), 4.0) / Float64(b * b)) - -1.0) * (b ^ 4.0));
	else
		tmp = fma(Float64(b * b), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 12.6], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[b, 5e+153], N[(N[(N[(N[(a * N[(a + a), $MachinePrecision] + 4.0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 12.5999999999999996

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if 12.5999999999999996 < b < 5.00000000000000018e153

   1. Initial program 99.7%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 95.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. sqr-pow N/A

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

      10. lower-pow.f64 95.5

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

   6. Applied rewrites 95.5%

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

   if 5.00000000000000018e153 < b

   1. Initial program 100.0%

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

   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. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 100.0%

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

Alternative 3: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 12.6:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(a, a + a, 4\right)}{b \cdot b} - -1\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 12.6)
   (fma (* a a) (* a a) -1.0)
   (if (<= b 5e+153)
     (* (- (/ (fma a (+ a a) 4.0) (* b b)) -1.0) (* (* (* b b) b) b))
     (fma (* b b) 4.0 -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 12.6) {
		tmp = fma((a * a), (a * a), -1.0);
	} else if (b <= 5e+153) {
		tmp = ((fma(a, (a + a), 4.0) / (b * b)) - -1.0) * (((b * b) * b) * b);
	} else {
		tmp = fma((b * b), 4.0, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 12.6)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	elseif (b <= 5e+153)
		tmp = Float64(Float64(Float64(fma(a, Float64(a + a), 4.0) / Float64(b * b)) - -1.0) * Float64(Float64(Float64(b * b) * b) * b));
	else
		tmp = fma(Float64(b * b), 4.0, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 12.6], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[b, 5e+153], N[(N[(N[(N[(a * N[(a + a), $MachinePrecision] + 4.0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 12.5999999999999996

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if 12.5999999999999996 < b < 5.00000000000000018e153

   1. Initial program 99.7%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 95.4%

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

   if 5.00000000000000018e153 < b

   1. Initial program 100.0%

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

   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. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 100.0%

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

Alternative 4: 81.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.6e+24) (fma (* a a) (* a a) -1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.6e+24) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.6e+24)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.6e+24], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.6000000000000003e24

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 78.5

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

   4. Applied rewrites 78.5%

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

   if 5.6000000000000003e24 < b

   1. Initial program 99.9%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 92.4

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

   4. Applied rewrites 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

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

      6. metadata-eval N/A

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

      7. lift-pow.f64 92.4

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

   6. Applied rewrites 92.4%

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

Alternative 5: 81.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.6e+24) (fma (* a a) (* a a) -1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.6e+24) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.6e+24)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.6e+24], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.6000000000000003e24

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0

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

      1. metadata-eval N/A

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

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

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 78.5

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

   4. Applied rewrites 78.5%

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

   if 5.6000000000000003e24 < b

   1. Initial program 99.9%

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

   2. Taylor expanded in b around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 92.4

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

   4. Applied rewrites 92.4%

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

Alternative 6: 66.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.8:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot a\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.8) (fma (* b b) 4.0 -1.0) (* (* (* a a) a) a)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.8) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = ((a * a) * a) * a;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.8)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = Float64(Float64(Float64(a * a) * a) * a);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.8], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.7999999999999998

   1. Initial program 99.9%

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

   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. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 79.3

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

   4. Applied rewrites 79.3%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 59.5%

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

   if 2.7999999999999998 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

Alternative 7: 66.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.8:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 4, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.8) (fma (* b b) 4.0 -1.0) (* (* a a) (* a a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.8) {
		tmp = fma((b * b), 4.0, -1.0);
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.8)
		tmp = fma(Float64(b * b), 4.0, -1.0);
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.8], N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.7999999999999998

   1. Initial program 99.9%

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

   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. metadata-eval N/A

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 79.3

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

   4. Applied rewrites 79.3%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 59.5%

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

   if 2.7999999999999998 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 87.9

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

   6. Applied rewrites 87.9%

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

Alternative 8: 51.2% accurate, 4.1× 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 99.9%

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

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. metadata-eval N/A

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lower-fma.f64 69.4

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

4. Applied rewrites 69.4%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 51.2%

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

Alternative 9: 25.2% accurate, 36.7× 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 99.9%

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

2. Taylor expanded in b around 0

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

   1. metadata-eval N/A

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

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

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 69.1

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

4. Applied rewrites 69.1%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 25.2%

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

Reproduce

herbie shell --seed 2025113 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))