Bouland and Aaronson, Equation (24)

Percentage Accurate: 73.1% → 98.2%

Time: 3.4s

Alternatives: 11

Speedup: 3.3×

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

Specification

Math

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

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: 73.1% 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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(3 + a\right)\\ t_1 := \mathsf{fma}\left(b, b, a \cdot a\right)\\ \mathbf{if}\;\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + t\_0\right)\right) - 1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, \mathsf{fma}\left(a \cdot a, 1 - a, t\_0\right) \cdot 4 - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* b b) (+ 3.0 a))) (t_1 (fma b b (* a a))))
   (if (<=
        (-
         (+
          (pow (+ (* a a) (* b b)) 2.0)
          (* 4.0 (+ (* (* a a) (- 1.0 a)) t_0)))
         1.0)
        INFINITY)
     (fma t_1 t_1 (- (* (fma (* a a) (- 1.0 a) t_0) 4.0) 1.0))
     (* (* a a) (* a a)))))

C

double code(double a, double b) {
	double t_0 = (b * b) * (3.0 + a);
	double t_1 = fma(b, b, (a * a));
	double tmp;
	if (((pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 - a)) + t_0))) - 1.0) <= ((double) INFINITY)) {
		tmp = fma(t_1, t_1, ((fma((a * a), (1.0 - a), t_0) * 4.0) - 1.0));
	} else {
		tmp = (a * a) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(Float64(b * b) * Float64(3.0 + a))
	t_1 = fma(b, b, Float64(a * a))
	tmp = 0.0
	if (Float64(Float64((Float64(Float64(a * a) + Float64(b * b)) ^ 2.0) + Float64(4.0 * Float64(Float64(Float64(a * a) * Float64(1.0 - a)) + t_0))) - 1.0) <= Inf)
		tmp = fma(t_1, t_1, Float64(Float64(fma(Float64(a * a), Float64(1.0 - a), t_0) * 4.0) - 1.0));
	else
		tmp = Float64(Float64(a * a) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(3.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * b + N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], Infinity], N[(t$95$1 * t$95$1 + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 - a), $MachinePrecision] + t$95$0), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64)) < +inf.0

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   if +inf.0 < (-.f64 (+.f64 (pow.f64 (+.f64 (*.f64 a a) (*.f64 b b)) #s(literal 2 binary64)) (*.f64 #s(literal 4 binary64) (+.f64 (*.f64 (*.f64 a a) (-.f64 #s(literal 1 binary64) a)) (*.f64 (*.f64 b b) (+.f64 #s(literal 3 binary64) a))))) #s(literal 1 binary64))

   1. Initial program 0.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 93.9

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

   4. Applied rewrites 93.9%

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

Alternative 2: 94.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(1 - a, \left(a \cdot a\right) \cdot 4, \left(\left(a \cdot a\right) \cdot a\right) \cdot a\right) - 1\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 12\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e-9)
   (- (fma (- 1.0 a) (* (* a a) 4.0) (* (* (* a a) a) a)) 1.0)
   (if (<= a 5.6e+26)
     (- (fma (* b b) (* b b) (* (* b b) 12.0)) 1.0)
     (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-9) {
		tmp = fma((1.0 - a), ((a * a) * 4.0), (((a * a) * a) * a)) - 1.0;
	} else if (a <= 5.6e+26) {
		tmp = fma((b * b), (b * b), ((b * b) * 12.0)) - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.5e-9)
		tmp = Float64(fma(Float64(1.0 - a), Float64(Float64(a * a) * 4.0), Float64(Float64(Float64(a * a) * a) * a)) - 1.0);
	elseif (a <= 5.6e+26)
		tmp = Float64(fma(Float64(b * b), Float64(b * b), Float64(Float64(b * b) * 12.0)) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.5e-9], N[(N[(N[(1.0 - a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 5.6e+26], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.5e-9

   1. Initial program 62.3%

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

   2. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. unpow3 N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. metadata-eval N/A

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

      19. pow-plus N/A

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

      20. lower-*.f64 N/A

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

      21. unpow3 N/A

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

      22. lower-*.f64 N/A

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

      23. lift-*.f64 88.7

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

   6. Applied rewrites 88.7%

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

   if -8.5e-9 < a < 5.59999999999999999e26

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if 5.59999999999999999e26 < a

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 93.1

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

   4. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow3 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 93.2

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

   6. Applied rewrites 93.2%

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

Alternative 3: 94.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-9}:\\ \;\;\;\;\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot b, \left(b \cdot b\right) \cdot 12\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{a}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e-9)
   (- (* (* (fma (- a 4.0) a 4.0) a) a) 1.0)
   (if (<= a 5.6e+26)
     (- (fma (* b b) (* b b) (* (* b b) 12.0)) 1.0)
     (pow a 4.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e-9) {
		tmp = ((fma((a - 4.0), a, 4.0) * a) * a) - 1.0;
	} else if (a <= 5.6e+26) {
		tmp = fma((b * b), (b * b), ((b * b) * 12.0)) - 1.0;
	} else {
		tmp = pow(a, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.5e-9)
		tmp = Float64(Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * a) * a) - 1.0);
	elseif (a <= 5.6e+26)
		tmp = Float64(fma(Float64(b * b), Float64(b * b), Float64(Float64(b * b) * 12.0)) - 1.0);
	else
		tmp = a ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.5e-9], N[(N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[a, 5.6e+26], N[(N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[a, 4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.5e-9

   1. Initial program 62.3%

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

   2. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 88.6

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

   7. Applied rewrites 88.6%

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

   9. Applied rewrites 88.7%

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

   if -8.5e-9 < a < 5.59999999999999999e26

   1. Initial program 99.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if 5.59999999999999999e26 < a

   1. Initial program 25.0%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 93.1

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

   4. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow3 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 93.2

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

   6. Applied rewrites 93.2%

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

Alternative 4: 82.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+27}:\\ \;\;\;\;\left(\mathsf{fma}\left(a - 4, a, 4\right) \cdot a\right) \cdot a - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.7e+27) (- (* (* (fma (- a 4.0) a 4.0) a) a) 1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.7e+27) {
		tmp = ((fma((a - 4.0), a, 4.0) * a) * a) - 1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.7e+27)
		tmp = Float64(Float64(Float64(fma(Float64(a - 4.0), a, 4.0) * a) * a) - 1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.7e+27], N[(N[(N[(N[(N[(a - 4.0), $MachinePrecision] * a + 4.0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6999999999999997e27

   1. Initial program 76.1%

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

   2. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 65.4

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 78.6

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

   7. Applied rewrites 78.6%

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

   9. Applied rewrites 78.7%

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

   if 2.6999999999999997e27 < b

   1. Initial program 63.2%

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

   2. 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. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 92.9

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

   4. Applied rewrites 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow3 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 93.0

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

   6. Applied rewrites 93.0%

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

Alternative 5: 82.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+27}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;{b}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.7e+27) (- (* (* a a) (* a a)) 1.0) (pow b 4.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.7e+27) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = pow(b, 4.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.7d+27) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = b ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.7e+27) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = Math.pow(b, 4.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.7e+27:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = math.pow(b, 4.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.7e+27)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = b ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.7e+27)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = b ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.7e+27], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[Power[b, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6999999999999997e27

   1. Initial program 76.1%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   if 2.6999999999999997e27 < b

   1. Initial program 63.2%

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

   2. 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. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 92.9

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

   4. Applied rewrites 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow3 N/A

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

      6. pow-plus N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 93.0

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

   6. Applied rewrites 93.0%

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

Alternative 6: 81.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+27}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.7e+27) (- (* (* a a) (* a a)) 1.0) (* (* (* b b) b) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.7e+27) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.7d+27) then
        tmp = ((a * a) * (a * a)) - 1.0d0
    else
        tmp = ((b * b) * b) * b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.7e+27) {
		tmp = ((a * a) * (a * a)) - 1.0;
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.7e+27:
		tmp = ((a * a) * (a * a)) - 1.0
	else:
		tmp = ((b * b) * b) * b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.7e+27)
		tmp = Float64(Float64(Float64(a * a) * Float64(a * a)) - 1.0);
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.7e+27)
		tmp = ((a * a) * (a * a)) - 1.0;
	else
		tmp = ((b * b) * b) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.7e+27], N[(N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6999999999999997e27

   1. Initial program 76.1%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   if 2.6999999999999997e27 < b

   1. Initial program 63.2%

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

   3. Applied rewrites 65.9%

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

   4. Taylor expanded in b around inf

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

      1. associate-+r- N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. pow2 N/A

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

      12. pow3 N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 92.9%

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

Alternative 7: 81.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-20}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+27}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.9e-20)
   (- (* (* a a) 4.0) 1.0)
   (if (<= b 2.7e+27) (* (* a a) (* a a)) (* (* (* b b) b) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-20) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b <= 2.7e+27) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.9d-20) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else if (b <= 2.7d+27) then
        tmp = (a * a) * (a * a)
    else
        tmp = ((b * b) * b) * b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-20) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b <= 2.7e+27) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = ((b * b) * b) * b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.9e-20:
		tmp = ((a * a) * 4.0) - 1.0
	elif b <= 2.7e+27:
		tmp = (a * a) * (a * a)
	else:
		tmp = ((b * b) * b) * b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.9e-20)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	elseif (b <= 2.7e+27)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(Float64(b * b) * b) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.9e-20)
		tmp = ((a * a) * 4.0) - 1.0;
	elseif (b <= 2.7e+27)
		tmp = (a * a) * (a * a);
	else
		tmp = ((b * b) * b) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.9e-20], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[b, 2.7e+27], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.9e-20

   1. Initial program 75.8%

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

   2. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 66.1

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 60.1

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

   7. Applied rewrites 60.1%

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

   if 2.9e-20 < b < 2.6999999999999997e27

   1. Initial program 81.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   if 2.6999999999999997e27 < b

   1. Initial program 63.2%

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

   3. Applied rewrites 65.9%

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

   4. Taylor expanded in b around inf

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

      1. associate-+r- N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. pow2 N/A

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

      12. pow3 N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 92.9%

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

Alternative 8: 67.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-20}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+27}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.9e-20)
   (- (* (* a a) 4.0) 1.0)
   (if (<= b 2.7e+27) (* (* a a) (* a a)) (* (* b b) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-20) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b <= 2.7e+27) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.9d-20) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else if (b <= 2.7d+27) then
        tmp = (a * a) * (a * a)
    else
        tmp = (b * b) * (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.9e-20) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else if (b <= 2.7e+27) {
		tmp = (a * a) * (a * a);
	} else {
		tmp = (b * b) * (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.9e-20:
		tmp = ((a * a) * 4.0) - 1.0
	elif b <= 2.7e+27:
		tmp = (a * a) * (a * a)
	else:
		tmp = (b * b) * (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.9e-20)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	elseif (b <= 2.7e+27)
		tmp = Float64(Float64(a * a) * Float64(a * a));
	else
		tmp = Float64(Float64(b * b) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.9e-20)
		tmp = ((a * a) * 4.0) - 1.0;
	elseif (b <= 2.7e+27)
		tmp = (a * a) * (a * a);
	else
		tmp = (b * b) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.9e-20], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[b, 2.7e+27], N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.9e-20

   1. Initial program 75.8%

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

   2. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 66.1

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 60.1

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

   7. Applied rewrites 60.1%

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

   if 2.9e-20 < b < 2.6999999999999997e27

   1. Initial program 81.9%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 46.4

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

   4. Applied rewrites 46.4%

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

   if 2.6999999999999997e27 < b

   1. Initial program 63.2%

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

   2. 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. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 92.9

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

   4. Applied rewrites 92.9%

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

Alternative 9: 67.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(a \cdot a\right)\\ \mathbf{if}\;a \leq -75000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 57000000:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) - 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) (* a a))))
   (if (<= a -75000.0)
     t_0
     (if (<= a 57000000.0) (- (* b (* b 12.0)) 1.0) t_0))))

C

double code(double a, double b) {
	double t_0 = (a * a) * (a * a);
	double tmp;
	if (a <= -75000.0) {
		tmp = t_0;
	} else if (a <= 57000000.0) {
		tmp = (b * (b * 12.0)) - 1.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (a * a)
    if (a <= (-75000.0d0)) then
        tmp = t_0
    else if (a <= 57000000.0d0) then
        tmp = (b * (b * 12.0d0)) - 1.0d0
    else
        tmp = t_0
    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 (a <= -75000.0) {
		tmp = t_0;
	} else if (a <= 57000000.0) {
		tmp = (b * (b * 12.0)) - 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * (a * a)
	tmp = 0
	if a <= -75000.0:
		tmp = t_0
	elif a <= 57000000.0:
		tmp = (b * (b * 12.0)) - 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * Float64(a * a))
	tmp = 0.0
	if (a <= -75000.0)
		tmp = t_0;
	elseif (a <= 57000000.0)
		tmp = Float64(Float64(b * Float64(b * 12.0)) - 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * (a * a);
	tmp = 0.0;
	if (a <= -75000.0)
		tmp = t_0;
	elseif (a <= 57000000.0)
		tmp = (b * (b * 12.0)) - 1.0;
	else
		tmp = t_0;
	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[a, -75000.0], t$95$0, If[LessEqual[a, 57000000.0], N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -75000 or 5.7e7 < a

   1. Initial program 45.5%

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

   2. Taylor expanded in a around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 90.0

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

   4. Applied rewrites 90.0%

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

   if -75000 < a < 5.7e7

   1. Initial program 99.7%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

      4. lift-*.f64 75.2

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

   7. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 75.2

         \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]

   9. Applied rewrites 75.2%

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

Alternative 10: 61.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{+153}:\\ \;\;\;\;\left(a \cdot a\right) \cdot 4 - 1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot 12\right) - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.9e+153) (- (* (* a a) 4.0) 1.0) (- (* b (* b 12.0)) 1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.9e+153) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (b * (b * 12.0)) - 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 <= 3.9d+153) then
        tmp = ((a * a) * 4.0d0) - 1.0d0
    else
        tmp = (b * (b * 12.0d0)) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.9e+153) {
		tmp = ((a * a) * 4.0) - 1.0;
	} else {
		tmp = (b * (b * 12.0)) - 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.9e+153:
		tmp = ((a * a) * 4.0) - 1.0
	else:
		tmp = (b * (b * 12.0)) - 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.9e+153)
		tmp = Float64(Float64(Float64(a * a) * 4.0) - 1.0);
	else
		tmp = Float64(Float64(b * Float64(b * 12.0)) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.9e+153)
		tmp = ((a * a) * 4.0) - 1.0;
	else
		tmp = (b * (b * 12.0)) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.9e+153], N[(N[(N[(a * a), $MachinePrecision] * 4.0), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(b * N[(b * 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.89999999999999983e153

   1. Initial program 75.2%

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

   2. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 61.0

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

   4. Applied rewrites 61.0%

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

   5. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 55.4

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

   7. Applied rewrites 55.4%

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

   if 3.89999999999999983e153 < b

   1. Initial program 59.0%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

      4. lift-*.f64 100.0

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot 12 - 1 \]

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 100.0

         \[\leadsto b \cdot \left(b \cdot 12\right) - 1 \]

   9. Applied rewrites 100.0%

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

Alternative 11: 51.7% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

2. Taylor expanded in b around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift--.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 56.4

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

4. Applied rewrites 56.4%

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

5. Taylor expanded in a around 0

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

   1. pow2 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lift-*.f64 51.7

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

7. Applied rewrites 51.7%

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

Reproduce

herbie shell --seed 2025120 
(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))