Bouland and Aaronson, Equation (26)

Percentage Accurate: 99.9% → 99.9%

Time: 2.9s

Alternatives: 10

Speedup: N/A×

• 60.7% 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, 1.2× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b b (* a a)))) (fma t_0 t_0 (fma (* b b) 4.0 -1.0))))

C

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

Julia

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

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * 4.0 + -1.0), $MachinePrecision]), $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate--l+ N/A

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

   10. unpow2 N/A

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

   11. lower-fma.f64 N/A

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

3. Applied rewrites 99.9%

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

Alternative 2: 83.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \left(b \cdot b\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a a (* b b))))
   (if (<= b 5.8e-8)
     (fma (* a a) (* a a) -1.0)
     (fma t_0 t_0 (* (* b b) 4.0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, a, (b * b));
	double tmp;
	if (b <= 5.8e-8) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma(t_0, t_0, ((b * b) * 4.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(a, a, Float64(b * b))
	tmp = 0.0
	if (b <= 5.8e-8)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(t_0, t_0, Float64(Float64(b * b) * 4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 5.8e-8], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.8000000000000003e-8

   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 78.8

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

   4. Applied rewrites 78.8%

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

   if 5.8000000000000003e-8 < 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate--l+ N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 98.6

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

   8. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 98.6

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

   10. Applied rewrites 98.6%

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

Alternative 3: 83.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 13.8:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a + a, 4\right)\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 13.8)
   (fma (* a a) (* a a) -1.0)
   (fma (* (fma b b (fma a (+ a a) 4.0)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 13.8) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma((fma(b, b, fma(a, (a + a), 4.0)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 13.8)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(fma(b, b, fma(a, Float64(a + a), 4.0)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 13.8], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + N[(a * N[(a + a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 13.800000000000001

   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 78.7

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

   4. Applied rewrites 78.7%

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

   if 13.800000000000001 < 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 a around 0

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

      1. metadata-eval N/A

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

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

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

   4. Applied rewrites 96.5%

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

Alternative 4: 83.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6800:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, \left(a + a\right) \cdot a\right) \cdot \left(b \cdot b\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6800.0)
   (fma (* a a) (* a a) -1.0)
   (+ (* (fma b b (* (+ a a) a)) (* b b)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6800.0) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = (fma(b, b, ((a + a) * a)) * (b * b)) + -1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6800.0)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(fma(b, b, Float64(Float64(a + a) * a)) * Float64(b * b)) + -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6800.0], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + N[(N[(a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6800

   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 78.6

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

   4. Applied rewrites 78.6%

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

   if 6800 < 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 a around 0

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in a around inf

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

      1. count-2-rev N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 96.6

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

   10. Applied rewrites 96.6%

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

       1. lift-fma.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. pow2 N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 96.5

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

   12. Applied rewrites 96.5%

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

Alternative 5: 83.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6800:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, b, \left(a + a\right) \cdot a\right) \cdot b, b, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6800.0)
   (fma (* a a) (* a a) -1.0)
   (fma (* (fma b b (* (+ a a) a)) b) b -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6800.0) {
		tmp = fma((a * a), (a * a), -1.0);
	} else {
		tmp = fma((fma(b, b, ((a + a) * a)) * b), b, -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6800.0)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = fma(Float64(fma(b, b, Float64(Float64(a + a) * a)) * b), b, -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6800.0], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(b * b + N[(N[(a + a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6800

   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 78.6

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

   4. Applied rewrites 78.6%

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

   if 6800 < 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 a around 0

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 96.7%

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

   8. Taylor expanded in a around inf

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

      1. count-2-rev N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 96.6

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

   10. Applied rewrites 96.6%

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

Alternative 6: 82.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 0.021:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b, 4\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 0.021) (fma (* b b) (fma b b 4.0) -1.0) (fma (* a a) (* a a) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= 0.021) {
		tmp = fma((b * b), fma(b, b, 4.0), -1.0);
	} else {
		tmp = fma((a * a), (a * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 0.021)
		tmp = fma(Float64(b * b), fma(b, b, 4.0), -1.0);
	else
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 0.021], N[(N[(b * b), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 0.0210000000000000013

   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 79.7

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

   4. Applied rewrites 79.7%

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

   if 0.0210000000000000013 < 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 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 88.9

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

   4. Applied rewrites 88.9%

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

Alternative 7: 81.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 750000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, a \cdot a, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 750000.0) (fma (* a a) (* a a) -1.0) (* (* b b) (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 750000.0) {
		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 <= 750000.0)
		tmp = fma(Float64(a * a), Float64(a * a), -1.0);
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 750000.0], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5e5

   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 78.6

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

   4. Applied rewrites 78.6%

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

   if 7.5e5 < 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 90.0

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

   4. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \]
   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. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 89.9

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

   6. Applied rewrites 89.9%

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

Alternative 8: 69.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;b \leq 2.45 \cdot 10^{-244}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;b \leq 9200000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= b 2.45e-244)
     t_0
     (if (<= b 1.45e-15)
       -1.0
       (if (<= b 9200000000.0) t_0 (* (* b b) (* b b)))))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (b <= 2.45e-244) {
		tmp = t_0;
	} else if (b <= 1.45e-15) {
		tmp = -1.0;
	} else if (b <= 9200000000.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (a * a)
    if (b <= 2.45d-244) then
        tmp = t_0
    else if (b <= 1.45d-15) then
        tmp = -1.0d0
    else if (b <= 9200000000.0d0) then
        tmp = t_0
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (b <= 2.45e-244) {
		tmp = t_0;
	} else if (b <= 1.45e-15) {
		tmp = -1.0;
	} else if (b <= 9200000000.0) {
		tmp = t_0;
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * a)
	tmp = 0
	if b <= 2.45e-244:
		tmp = t_0
	elif b <= 1.45e-15:
		tmp = -1.0
	elif b <= 9200000000.0:
		tmp = t_0
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (b <= 2.45e-244)
		tmp = t_0;
	elseif (b <= 1.45e-15)
		tmp = -1.0;
	elseif (b <= 9200000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * (a * a);
	tmp = 0.0;
	if (b <= 2.45e-244)
		tmp = t_0;
	elseif (b <= 1.45e-15)
		tmp = -1.0;
	elseif (b <= 9200000000.0)
		tmp = t_0;
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.45e-244], t$95$0, If[LessEqual[b, 1.45e-15], -1.0, If[LessEqual[b, 9200000000.0], t$95$0, N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.45000000000000007e-244 or 1.45000000000000009e-15 < b < 9.2e9

   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 45.4

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

   4. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a} \]
   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 45.4

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

   6. Applied rewrites 45.4%

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

   if 2.45000000000000007e-244 < b < 1.45000000000000009e-15

   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 47.7%

      \[\leadsto -1 \]

   if 9.2e9 < 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 41.0

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

   4. Applied rewrites 41.0%

      \[\leadsto \color{blue}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \]
   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. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 40.9

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

   6. Applied rewrites 40.9%

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

Alternative 9: 56.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], -0.5], -1.0, N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 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 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 98.5%

      \[\leadsto -1 \]

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

   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 59.7

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

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot a\right) \cdot a} \]
   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 59.7

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

   6. Applied rewrites 59.7%

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

Alternative 10: 24.8% 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 69.2

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

4. Applied rewrites 69.2%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 24.8%

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

Reproduce

herbie shell --seed 2025114 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))