Bouland and Aaronson, Equation (24)

Percentage Accurate: 74.0% → 99.8%

Time: 7.7s

Alternatives: 10

Speedup: 6.2×

• 60.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 2, \mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right)\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(-4, a, 4\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (fma
   (* (fma (* a a) 2.0 (fma b b (fma 4.0 a 12.0))) b)
   b
   (* (* a a) (fma a a (fma -4.0 a 4.0))))
  1.0))

C

double code(double a, double b) {
	return fma((fma((a * a), 2.0, fma(b, b, fma(4.0, a, 12.0))) * b), b, ((a * a) * fma(a, a, fma(-4.0, a, 4.0)))) - 1.0;
}

Julia

function code(a, b)
	return Float64(fma(Float64(fma(Float64(a * a), 2.0, fma(b, b, fma(4.0, a, 12.0))) * b), b, Float64(Float64(a * a) * fma(a, a, fma(-4.0, a, 4.0)))) - 1.0)
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 2.0 + N[(b * b + N[(4.0 * a + 12.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(a * a + N[(-4.0 * a + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 2, \mathsf{fma}\left(b, b, \mathsf{fma}\left(4, a, 12\right)\right)\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(-4, a, 4\right)\right)\right)} - 1 \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, a, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right) \cdot b\right) \cdot b\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (fma
   (* (fma (- a 4.0) a 4.0) a)
   a
   (* (* (fma (fma 2.0 a 4.0) a (fma b b 12.0)) b) b))
  1.0))

C

double code(double a, double b) {
	return fma((fma((a - 4.0), a, 4.0) * a), a, ((fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)) * b) * b)) - 1.0;
}

Julia

function code(a, b)
	return Float64(fma(Float64(fma(Float64(a - 4.0), a, 4.0) * a), a, Float64(Float64(fma(fma(2.0, a, 4.0), a, fma(b, b, 12.0)) * b) * b)) - 1.0)
end

Wolfram

code[a_, b_] := N[(N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(N[(2.0 * a + 4.0), $MachinePrecision] * a + N[(b * b + 12.0), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a, \color{blue}{a}, \left(\mathsf{fma}\left(\mathsf{fma}\left(2, a, 4\right), a, \mathsf{fma}\left(b, b, 12\right)\right) \cdot b\right) \cdot b\right) - 1 \]
9. Add Preprocessing

Alternative 3: 99.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(-4, a, 4\right)\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (fma (* (fma b b 12.0) b) b (* (* a a) (fma a a (fma -4.0 a 4.0)))) 1.0))

C

double code(double a, double b) {
	return fma((fma(b, b, 12.0) * b), b, ((a * a) * fma(a, a, fma(-4.0, a, 4.0)))) - 1.0;
}

Julia

function code(a, b)
	return Float64(fma(Float64(fma(b, b, 12.0) * b), b, Float64(Float64(a * a) * fma(a, a, fma(-4.0, a, 4.0)))) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(-4, a, 4\right)\right)\right) - 1 \]
9. Add Preprocessing

Alternative 4: 99.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(a, a + -4, 4\right), a \cdot a, \mathsf{fma}\left(b, b, 12\right) \cdot \left(b \cdot b\right)\right) - 1 \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (- (fma (fma a (+ a -4.0) 4.0) (* a a) (* (fma b b 12.0) (* b b))) 1.0))

C

double code(double a, double b) {
	return fma(fma(a, (a + -4.0), 4.0), (a * a), (fma(b, b, 12.0) * (b * b))) - 1.0;
}

Julia

function code(a, b)
	return Float64(fma(fma(a, Float64(a + -4.0), 4.0), Float64(a * a), Float64(fma(b, b, 12.0) * Float64(b * b))) - 1.0)
end

Wolfram

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

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 99.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, b, 12\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(a, a, \mathsf{fma}\left(-4, a, 4\right)\right)\right) - 1 \]
9. Step-by-step derivation

10. Applied rewrites 99.5%

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

Alternative 5: 92.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+76} \lor \neg \left(a \leq 1.05 \cdot 10^{+71}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= a -3.8e+76) (not (<= a 1.05e+71)))
   (- (* (* a a) (* a a)) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((a <= -3.8e+76) || !(a <= 1.05e+71)) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if ((a <= -3.8e+76) || !(a <= 1.05e+71))
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[Or[LessEqual[a, -3.8e+76], N[Not[LessEqual[a, 1.05e+71]], $MachinePrecision]], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.80000000000000024e76 or 1.04999999999999995e71 < a

   1. Initial program 31.7%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   if -3.80000000000000024e76 < a < 1.04999999999999995e71

   1. Initial program 95.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 92.4%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+76} \lor \neg \left(a \leq 1.05 \cdot 10^{+71}\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 640000:\\ \;\;\;\;\mathsf{fma}\left(a, a + -4, 4\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 640000.0)
   (- (* (fma a (+ a -4.0) 4.0) (* a a)) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 640000.0) {
		tmp = (fma(a, (a + -4.0), 4.0) * (a * a)) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 640000.0)
		tmp = Float64(Float64(fma(a, Float64(a + -4.0), 4.0) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 640000.0], N[(N[(N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4e5

   1. Initial program 71.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 76.9%

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

   if 6.4e5 < b

   1. Initial program 64.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 86.3%

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

Alternative 7: 81.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 640000:\\ \;\;\;\;\left(\mathsf{fma}\left(a, a + -4, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(b, b, 12\right) \cdot b\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 640000.0)
   (- (* (* (fma a (+ a -4.0) 4.0) a) a) 1.0)
   (- (* (* (fma b b 12.0) b) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 640000.0) {
		tmp = ((fma(a, (a + -4.0), 4.0) * a) * a) - 1.0;
	} else {
		tmp = ((fma(b, b, 12.0) * b) * b) - 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 640000.0)
		tmp = Float64(Float64(Float64(fma(a, Float64(a + -4.0), 4.0) * a) * a) - 1.0);
	else
		tmp = Float64(Float64(Float64(fma(b, b, 12.0) * b) * b) - 1.0);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 640000.0], N[(N[(N[(N[(a * N[(a + -4.0), $MachinePrecision] + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(b * b + 12.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4e5

   1. Initial program 71.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 76.9%

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

   if 6.4e5 < b

   1. Initial program 64.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 86.3%

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

Alternative 8: 75.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+151}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 12\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8e+151) (- (* (* a a) (* a a)) 1.0) (- (* (* b 12.0) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8e+151) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b * 12.0) * b) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 8d+151) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = ((b * 12.0d0) * b) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8e+151) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b * 12.0) * b) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8e+151:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = ((b * 12.0) * b) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8e+151)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * 12.0) * b) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8e+151)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = ((b * 12.0) * b) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8e+151], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * 12.0), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.00000000000000014e151

   1. Initial program 71.2%

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

   3. Taylor expanded in a around inf

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

      1. lower-pow.f64 70.4

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

   5. Applied rewrites 70.4%

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

   7. Applied rewrites 70.4%

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

   if 8.00000000000000014e151 < b

   1. Initial program 57.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \left(b \cdot 12\right) \cdot b - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{+151}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 12\right) \cdot b - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.6e+151) (- (* (* a a) 4.0) 1.0) (- (* (* b 12.0) b) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.6e+151) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * 12.0) * b) - 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 7.6d+151) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = ((b * 12.0d0) * b) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7.6e+151) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = ((b * 12.0) * b) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7.6e+151:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = ((b * 12.0) * b) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.6e+151)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * 12.0) * b) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7.6e+151)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = ((b * 12.0) * b) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.6e+151], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * 12.0), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000001e151

   1. Initial program 71.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. mul-1-neg N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. distribute-lft-neg-in N/A

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

      14. metadata-eval N/A

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

      15. distribute-rgt-out N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-fma.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-fma.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 50.7%

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

   if 7.6000000000000001e151 < b

   1. Initial program 57.7%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \left(b \cdot 12\right) \cdot b - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.0% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \left(b \cdot 12\right) \cdot b - 1 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* b 12.0) b) 1.0))

C

double code(double a, double b) {
	return ((b * 12.0) * 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 = ((b * 12.0d0) * b) - 1.0d0
end function

Java

public static double code(double a, double b) {
	return ((b * 12.0) * b) - 1.0;
}

Python

def code(a, b):
	return ((b * 12.0) * b) - 1.0

Julia

function code(a, b)
	return Float64(Float64(Float64(b * 12.0) * b) - 1.0)
end

MATLAB

function tmp = code(a, b)
	tmp = ((b * 12.0) * b) - 1.0;
end

Wolfram

code[a_, b_] := N[(N[(N[(b * 12.0), $MachinePrecision] * b), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 67.4

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

5. Applied rewrites 67.4%

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

6. Taylor expanded in b around 0

   \[\leadsto 12 \cdot \color{blue}{{b}^{2}} - 1 \]
7. Step-by-step derivation

8. Applied rewrites 49.4%

   \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{12} - 1 \]
9. Step-by-step derivation

10. Applied rewrites 49.4%

    \[\leadsto \left(b \cdot 12\right) \cdot b - 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))