Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A

Percentage Accurate: 90.0% → 96.5%

Time: 4.7s

Alternatives: 14

Speedup: 0.5×

• 47.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.0% accurate, 1.0× speedup

Math

\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = 2.0d0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

Python

def code(x, y, z, t, a, b, c, i):
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+283}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, \left(-\left(-\left(-c\right)\right)\right) \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+206}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, \left(\left(-c\right) \cdot i\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ a (* b c)) c)))
   (if (<= t_1 -2e+283)
     (* 2.0 (fma y x (* (- (- (- c))) (* i (fma b c a)))))
     (if (<= t_1 5e+206)
       (* 2.0 (fma (* (fma c b a) c) (- i) (fma t z (* y x))))
       (* 2.0 (fma y x (* (* (- c) i) (fma b c a))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a + (b * c)) * c;
	double tmp;
	if (t_1 <= -2e+283) {
		tmp = 2.0 * fma(y, x, (-(-(-c)) * (i * fma(b, c, a))));
	} else if (t_1 <= 5e+206) {
		tmp = 2.0 * fma((fma(c, b, a) * c), -i, fma(t, z, (y * x)));
	} else {
		tmp = 2.0 * fma(y, x, ((-c * i) * fma(b, c, a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_1 <= -2e+283)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(-Float64(-Float64(-c))) * Float64(i * fma(b, c, a)))));
	elseif (t_1 <= 5e+206)
		tmp = Float64(2.0 * fma(Float64(fma(c, b, a) * c), Float64(-i), fma(t, z, Float64(y * x))));
	else
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(Float64(-c) * i) * fma(b, c, a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+283], N[(2.0 * N[(y * x + N[((-(-(-c))) * N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+206], N[(2.0 * N[(N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision] * (-i) + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(y * x + N[(N[((-c) * i), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -1.99999999999999991e283

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      6. lower-*.f64 70.3

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

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

         \[\leadsto 2 \cdot \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-c\right) \cdot \left(\color{blue}{i} \cdot \left(a + b \cdot c\right)\right)\right) \]

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

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{\left(\mathsf{neg}\left(\left(-c\right)\right)\right) \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

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

         \[\leadsto 2 \cdot \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-c\right)\right)\right)\right)\right) \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-c\right)\right)\right)\right)\right)} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-c\right)\right)\right)\right)\right)} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(y, \color{blue}{x}, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-c\right)\right)\right)\right)\right) \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(-c\right)\right)\right)\right)\right) \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-\left(\mathsf{neg}\left(\left(-c\right)\right)\right)\right) \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      12. lower-neg.f64 71.5

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-\left(-\left(-c\right)\right)\right) \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      13. lift-+.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-\left(-\left(-c\right)\right)\right) \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-\left(-\left(-c\right)\right)\right) \cdot \left(i \cdot \left(b \cdot c + a\right)\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-\left(-\left(-c\right)\right)\right) \cdot \left(i \cdot \left(b \cdot c + a\right)\right)\right) \]

      16. lower-fma.f64 71.5

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(-\left(-\left(-c\right)\right)\right) \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right) \]

   6. Applied rewrites 71.5%

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, \color{blue}{x}, \left(-\left(-\left(-c\right)\right)\right) \cdot \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right)\right) \]

   if -1.99999999999999991e283 < (*.f64 (+.f64 a (*.f64 b c)) c) < 5.0000000000000002e206

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right) \]

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

         \[\leadsto 2 \cdot \color{blue}{\left(\left(x \cdot y + z \cdot t\right) + \left(\mathsf{neg}\left(\left(a + b \cdot c\right) \cdot c\right)\right) \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(a + b \cdot c\right) \cdot c\right)\right) \cdot i + \left(x \cdot y + z \cdot t\right)\right)} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)} + \left(x \cdot y + z \cdot t\right)\right) \]

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

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot \left(\mathsf{neg}\left(i\right)\right)} + \left(x \cdot y + z \cdot t\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(\left(a + b \cdot c\right) \cdot c, \mathsf{neg}\left(i\right), x \cdot y + z \cdot t\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{\left(a + b \cdot c\right)} \cdot c, \mathsf{neg}\left(i\right), x \cdot y + z \cdot t\right) \]

      9. +-commutative N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{\left(b \cdot c + a\right)} \cdot c, \mathsf{neg}\left(i\right), x \cdot y + z \cdot t\right) \]

      10. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\left(\color{blue}{b \cdot c} + a\right) \cdot c, \mathsf{neg}\left(i\right), x \cdot y + z \cdot t\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\left(\color{blue}{c \cdot b} + a\right) \cdot c, \mathsf{neg}\left(i\right), x \cdot y + z \cdot t\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(c, b, a\right)} \cdot c, \mathsf{neg}\left(i\right), x \cdot y + z \cdot t\right) \]

      13. lower-neg.f64 90.6

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, \color{blue}{-i}, x \cdot y + z \cdot t\right) \]

      14. lift-+.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \color{blue}{x \cdot y + z \cdot t}\right) \]

      15. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \color{blue}{z \cdot t + x \cdot y}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \color{blue}{z \cdot t} + x \cdot y\right) \]

      17. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \color{blue}{t \cdot z} + x \cdot y\right) \]

      18. lower-fma.f64 90.9

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}\right) \]

      19. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right)\right) \]

      20. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

      21. lower-*.f64 90.9

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

   3. Applied rewrites 90.9%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right) \cdot c, -i, \mathsf{fma}\left(t, z, y \cdot x\right)\right)} \]

   if 5.0000000000000002e206 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      6. lower-*.f64 70.3

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(a + b \cdot c\right) \cdot \color{blue}{i}\right)\right) \]

      4. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(b \cdot c + a\right) \cdot i\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(b \cdot c + a\right) \cdot i\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(c \cdot b + a\right) \cdot i\right)\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \left(c \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot \color{blue}{i}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \]

      12. lower-*.f64 69.6

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]

      13. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\left(c \cdot b + a\right) \cdot c\right) \cdot i\right) \]

      14. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\right) \]

      15. lower-fma.f64 69.6

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \]

   6. Applied rewrites 69.6%

      \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - i \cdot \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)}\right) \]

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

         \[\leadsto 2 \cdot \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)}\right) \]

      5. lift-neg.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot c\right)\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(b \cdot c + a\right) \cdot c\right)\right) \]

      7. add-flip N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(b \cdot c - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot c\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(c \cdot b - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot c\right)\right) \]

      9. add-flip N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\right) \]

      10. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y + \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \color{blue}{\left(-i\right)}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y + \color{blue}{\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)} \cdot \left(-i\right)\right) \]

      13. *-commutative N/A

          \[\leadsto 2 \cdot \left(y \cdot x + \color{blue}{\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)} \cdot \left(-i\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, \color{blue}{x}, \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(c, b, a\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      17. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot b + a\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      18. add-flip N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot b - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(b \cdot c - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      20. add-flip N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(b \cdot c + a\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      21. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(a + b \cdot c\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot \left(-i\right)\right) \cdot \left(a + b \cdot c\right)\right) \]

      23. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot \left(-i\right)\right) \cdot \left(a + b \cdot c\right)\right) \]

   8. Applied rewrites 74.0%

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

Alternative 2: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \left(-c\right) \cdot i\\ \mathbf{if}\;2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \leq \infty:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), t\_1, \mathsf{fma}\left(t, z, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t\_1 \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (- c) i)))
   (if (<= (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))) INFINITY)
     (* 2.0 (fma (fma c b a) t_1 (fma t z (* y x))))
     (* 2.0 (fma y x (* t_1 (fma b c a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -c * i;
	double tmp;
	if ((2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i))) <= ((double) INFINITY)) {
		tmp = 2.0 * fma(fma(c, b, a), t_1, fma(t, z, (y * x)));
	} else {
		tmp = 2.0 * fma(y, x, (t_1 * fma(b, c, a)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(-c) * i)
	tmp = 0.0
	if (Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i))) <= Inf)
		tmp = Float64(2.0 * fma(fma(c, b, a), t_1, fma(t, z, Float64(y * x))));
	else
		tmp = Float64(2.0 * fma(y, x, Float64(t_1 * fma(b, c, a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[((-c) * i), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 * N[(N[(c * b + a), $MachinePrecision] * t$95$1 + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(y * x + N[(t$95$1 * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i))) < +inf.0

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i}\right) \]

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

         \[\leadsto 2 \cdot \color{blue}{\left(\left(x \cdot y + z \cdot t\right) + \left(\mathsf{neg}\left(\left(a + b \cdot c\right) \cdot c\right)\right) \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(a + b \cdot c\right) \cdot c\right)\right) \cdot i + \left(x \cdot y + z \cdot t\right)\right)} \]

      5. distribute-lft-neg-out N/A

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right)\right)} + \left(x \cdot y + z \cdot t\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot i\right)\right) + \left(x \cdot y + z \cdot t\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto 2 \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(a + b \cdot c\right) \cdot \left(c \cdot i\right)}\right)\right) + \left(x \cdot y + z \cdot t\right)\right) \]

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

         \[\leadsto 2 \cdot \left(\color{blue}{\left(a + b \cdot c\right) \cdot \left(\mathsf{neg}\left(c \cdot i\right)\right)} + \left(x \cdot y + z \cdot t\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(a + b \cdot c, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{a + b \cdot c}, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      11. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{b \cdot c + a}, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      12. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{b \cdot c} + a, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      13. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{c \cdot b} + a, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(c, b, a\right)}, \mathsf{neg}\left(c \cdot i\right), x \cdot y + z \cdot t\right) \]

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

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot i}, x \cdot y + z \cdot t\right) \]

      16. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot i}, x \cdot y + z \cdot t\right) \]

      17. lower-neg.f64 94.9

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \color{blue}{\left(-c\right)} \cdot i, x \cdot y + z \cdot t\right) \]

      18. lift-+.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \color{blue}{x \cdot y + z \cdot t}\right) \]

      19. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \color{blue}{z \cdot t + x \cdot y}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \color{blue}{z \cdot t} + x \cdot y\right) \]

      21. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \color{blue}{t \cdot z} + x \cdot y\right) \]

      22. lower-fma.f64 95.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}\right) \]

      23. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \mathsf{fma}\left(t, z, \color{blue}{x \cdot y}\right)\right) \]

      24. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

      25. lower-*.f64 95.2

          \[\leadsto 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \mathsf{fma}\left(t, z, \color{blue}{y \cdot x}\right)\right) \]

   3. Applied rewrites 95.2%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), \left(-c\right) \cdot i, \mathsf{fma}\left(t, z, y \cdot x\right)\right)} \]

   if +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (+.f64 (*.f64 x y) (*.f64 z t)) (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)))

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      6. lower-*.f64 70.3

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(a + b \cdot c\right) \cdot \color{blue}{i}\right)\right) \]

      4. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(b \cdot c + a\right) \cdot i\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(b \cdot c + a\right) \cdot i\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(c \cdot b + a\right) \cdot i\right)\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \left(c \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot \color{blue}{i}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \]

      12. lower-*.f64 69.6

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]

      13. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\left(c \cdot b + a\right) \cdot c\right) \cdot i\right) \]

      14. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\right) \]

      15. lower-fma.f64 69.6

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \]

   6. Applied rewrites 69.6%

      \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - i \cdot \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)}\right) \]

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

         \[\leadsto 2 \cdot \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)}\right) \]

      5. lift-neg.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot c\right)\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(b \cdot c + a\right) \cdot c\right)\right) \]

      7. add-flip N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(b \cdot c - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot c\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(c \cdot b - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot c\right)\right) \]

      9. add-flip N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\right) \]

      10. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y + \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \color{blue}{\left(-i\right)}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y + \color{blue}{\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)} \cdot \left(-i\right)\right) \]

      13. *-commutative N/A

          \[\leadsto 2 \cdot \left(y \cdot x + \color{blue}{\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)} \cdot \left(-i\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, \color{blue}{x}, \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(c, b, a\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      17. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot b + a\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      18. add-flip N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot b - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(b \cdot c - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      20. add-flip N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(b \cdot c + a\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      21. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(a + b \cdot c\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot \left(-i\right)\right) \cdot \left(a + b \cdot c\right)\right) \]

      23. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot \left(-i\right)\right) \cdot \left(a + b \cdot c\right)\right) \]

   8. Applied rewrites 74.0%

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

Alternative 3: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{+114}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, \left(\left(-c\right) \cdot i\right) \cdot \mathsf{fma}\left(b, c, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (- (fma t z (* x y)) (* a (* c i))))))
   (if (<= (* z t) -1e+56)
     t_1
     (if (<= (* z t) 1e+114)
       (* 2.0 (fma y x (* (* (- c) i) (fma b c a))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (fma(t, z, (x * y)) - (a * (c * i)));
	double tmp;
	if ((z * t) <= -1e+56) {
		tmp = t_1;
	} else if ((z * t) <= 1e+114) {
		tmp = 2.0 * fma(y, x, ((-c * i) * fma(b, c, a)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * Float64(fma(t, z, Float64(x * y)) - Float64(a * Float64(c * i))))
	tmp = 0.0
	if (Float64(z * t) <= -1e+56)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e+114)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(Float64(-c) * i) * fma(b, c, a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+56], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e+114], N[(2.0 * N[(y * x + N[(N[((-c) * i), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000009e56 or 1e114 < (*.f64 z t)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in b around 0

      \[\leadsto 2 \cdot \color{blue}{\left(\left(t \cdot z + x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(\left(t \cdot z + x \cdot y\right) - \color{blue}{a \cdot \left(c \cdot i\right)}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - \color{blue}{a} \cdot \left(c \cdot i\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \color{blue}{\left(c \cdot i\right)}\right) \]

      5. lower-*.f64 73.7

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot \color{blue}{i}\right)\right) \]

   4. Applied rewrites 73.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)} \]

   if -1.00000000000000009e56 < (*.f64 z t) < 1e114

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      6. lower-*.f64 70.3

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(a + b \cdot c\right) \cdot \color{blue}{i}\right)\right) \]

      4. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(b \cdot c + a\right) \cdot i\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(b \cdot c + a\right) \cdot i\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(c \cdot b + a\right) \cdot i\right)\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \left(c \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot \color{blue}{i}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \]

      12. lower-*.f64 69.6

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]

      13. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\left(c \cdot b + a\right) \cdot c\right) \cdot i\right) \]

      14. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\right) \]

      15. lower-fma.f64 69.6

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \]

   6. Applied rewrites 69.6%

      \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - i \cdot \color{blue}{\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)}\right) \]

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

         \[\leadsto 2 \cdot \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(i\right)\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)}\right) \]

      5. lift-neg.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\color{blue}{\mathsf{fma}\left(b, c, a\right)} \cdot c\right)\right) \]

      6. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(b \cdot c + a\right) \cdot c\right)\right) \]

      7. add-flip N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(b \cdot c - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot c\right)\right) \]

      8. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(c \cdot b - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot c\right)\right) \]

      9. add-flip N/A

         \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\left(c \cdot b + a\right) \cdot c\right)\right) \]

      10. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y + \left(-i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right) \]

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y + \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \color{blue}{\left(-i\right)}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y + \color{blue}{\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)} \cdot \left(-i\right)\right) \]

      13. *-commutative N/A

          \[\leadsto 2 \cdot \left(y \cdot x + \color{blue}{\left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)} \cdot \left(-i\right)\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, \color{blue}{x}, \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \]

      15. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \left(-i\right)\right) \]

      16. associate-*l* N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(c, b, a\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      17. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot b + a\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      18. add-flip N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot b - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(b \cdot c - \left(\mathsf{neg}\left(a\right)\right)\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      20. add-flip N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(b \cdot c + a\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      21. +-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(a + b \cdot c\right) \cdot \left(c \cdot \left(-i\right)\right)\right) \]

      22. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot \left(-i\right)\right) \cdot \left(a + b \cdot c\right)\right) \]

      23. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \left(c \cdot \left(-i\right)\right) \cdot \left(a + b \cdot c\right)\right) \]

   8. Applied rewrites 74.0%

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

Alternative 4: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+108}:\\ \;\;\;\;\left(x \cdot y - \left(i \cdot c\right) \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (- (fma t z (* x y)) (* a (* c i))))))
   (if (<= (* z t) -1e+56)
     t_1
     (if (<= (* z t) 5e+108)
       (* (- (* x y) (* (* i c) (fma b c a))) 2.0)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 2.0 * (fma(t, z, (x * y)) - (a * (c * i)));
	double tmp;
	if ((z * t) <= -1e+56) {
		tmp = t_1;
	} else if ((z * t) <= 5e+108) {
		tmp = ((x * y) - ((i * c) * fma(b, c, a))) * 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(2.0 * Float64(fma(t, z, Float64(x * y)) - Float64(a * Float64(c * i))))
	tmp = 0.0
	if (Float64(z * t) <= -1e+56)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e+108)
		tmp = Float64(Float64(Float64(x * y) - Float64(Float64(i * c) * fma(b, c, a))) * 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(2.0 * N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+56], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e+108], N[(N[(N[(x * y), $MachinePrecision] - N[(N[(i * c), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000009e56 or 4.99999999999999991e108 < (*.f64 z t)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in b around 0

      \[\leadsto 2 \cdot \color{blue}{\left(\left(t \cdot z + x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(\left(t \cdot z + x \cdot y\right) - \color{blue}{a \cdot \left(c \cdot i\right)}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - \color{blue}{a} \cdot \left(c \cdot i\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \color{blue}{\left(c \cdot i\right)}\right) \]

      5. lower-*.f64 73.7

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot \color{blue}{i}\right)\right) \]

   4. Applied rewrites 73.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)} \]

   if -1.00000000000000009e56 < (*.f64 z t) < 4.99999999999999991e108

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      6. lower-*.f64 70.3

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2} \]

      3. lower-*.f64 70.3

         \[\leadsto \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \cdot 2} \]

   6. Applied rewrites 72.8%

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

Alternative 5: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+240}:\\ \;\;\;\;\mathsf{fma}\left(c, b, a\right) \cdot \left(\left(i \cdot c\right) \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+247}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -5e+240)
     (* (fma c b a) (* (* i c) -2.0))
     (if (<= t_1 2e+247)
       (* 2.0 (- (fma t z (* x y)) (* a (* c i))))
       (* (* -2.0 i) (* (fma c b a) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -5e+240) {
		tmp = fma(c, b, a) * ((i * c) * -2.0);
	} else if (t_1 <= 2e+247) {
		tmp = 2.0 * (fma(t, z, (x * y)) - (a * (c * i)));
	} else {
		tmp = (-2.0 * i) * (fma(c, b, a) * c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -5e+240)
		tmp = Float64(fma(c, b, a) * Float64(Float64(i * c) * -2.0));
	elseif (t_1 <= 2e+247)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(x * y)) - Float64(a * Float64(c * i))));
	else
		tmp = Float64(Float64(-2.0 * i) * Float64(fma(c, b, a) * c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+240], N[(N[(c * b + a), $MachinePrecision] * N[(N[(i * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+247], N[(2.0 * N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * i), $MachinePrecision] * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000003e240

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      5. lower-*.f64 47.1

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(a + b \cdot c\right) \cdot \color{blue}{i}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(a + b \cdot c\right) \cdot i\right) \]

      7. +-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(b \cdot c + a\right) \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(b \cdot c + a\right) \cdot i\right) \]

      9. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right) \]

      10. associate-*r* N/A

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

      11. associate-*l* N/A

          \[\leadsto \left(-2 \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right) \cdot i \]

      12. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\right) \cdot i \]

      13. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\right) \cdot i \]

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \cdot -2 \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \cdot -2 \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot i\right)\right) \cdot -2 \]

      20. associate-*l* N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 48.6%

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

   if -5.0000000000000003e240 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999999e247

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in b around 0

      \[\leadsto 2 \cdot \color{blue}{\left(\left(t \cdot z + x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(\left(t \cdot z + x \cdot y\right) - \color{blue}{a \cdot \left(c \cdot i\right)}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - \color{blue}{a} \cdot \left(c \cdot i\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \color{blue}{\left(c \cdot i\right)}\right) \]

      5. lower-*.f64 73.7

         \[\leadsto 2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot \color{blue}{i}\right)\right) \]

   4. Applied rewrites 73.7%

      \[\leadsto 2 \cdot \color{blue}{\left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)} \]

   if 1.9999999999999999e247 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      5. lower-*.f64 47.1

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

   6. Applied rewrites 46.8%

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

Alternative 6: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ t_2 := 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot \left(-2 \cdot c\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-81}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i))
        (t_2 (* 2.0 (- (* x y) (* (* (fma b c a) c) i)))))
   (if (<= t_1 (- INFINITY))
     (* (* (fma c b a) i) (* -2.0 c))
     (if (<= t_1 -2e-55)
       t_2
       (if (<= t_1 1e-81) (* 2.0 (fma t z (* x y))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double t_2 = 2.0 * ((x * y) - ((fma(b, c, a) * c) * i));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(c, b, a) * i) * (-2.0 * c);
	} else if (t_1 <= -2e-55) {
		tmp = t_2;
	} else if (t_1 <= 1e-81) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	t_2 = Float64(2.0 * Float64(Float64(x * y) - Float64(Float64(fma(b, c, a) * c) * i)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(c, b, a) * i) * Float64(-2.0 * c));
	elseif (t_1 <= -2e-55)
		tmp = t_2;
	elseif (t_1 <= 1e-81)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(x * y), $MachinePrecision] - N[(N[(N[(b * c + a), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * N[(-2.0 * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-55], t$95$2, If[LessEqual[t$95$1, 1e-81], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      5. lower-*.f64 47.1

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{\left(-2 \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{\left(-2 \cdot c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \left(\color{blue}{-2} \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(a + b \cdot c\right) \cdot i\right) \cdot \left(\color{blue}{-2} \cdot c\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(a + b \cdot c\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\left(b \cdot c + a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot c + a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      12. lower-*.f64 N/A

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

      13. lift-fma.f64 N/A

          \[\leadsto \left(\left(b \cdot c + a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\left(c \cdot b + a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      16. lower-*.f64 47.1

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

   6. Applied rewrites 47.1%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.99999999999999999e-55 or 9.9999999999999996e-82 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \color{blue}{c} \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      6. lower-*.f64 70.3

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 70.3%

      \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y - c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(a + b \cdot c\right) \cdot \color{blue}{i}\right)\right) \]

      4. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(a + b \cdot c\right) \cdot i\right)\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(b \cdot c + a\right) \cdot i\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(b \cdot c + a\right) \cdot i\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\left(c \cdot b + a\right) \cdot i\right)\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot y - c \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto 2 \cdot \left(x \cdot y - \left(c \cdot \mathsf{fma}\left(c, b, a\right)\right) \cdot \color{blue}{i}\right) \]

      10. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \]

      11. lift-*.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot i\right) \]

      12. lower-*.f64 69.6

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]

      13. lift-fma.f64 N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\left(c \cdot b + a\right) \cdot c\right) \cdot i\right) \]

      14. *-commutative N/A

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\left(b \cdot c + a\right) \cdot c\right) \cdot i\right) \]

      15. lower-fma.f64 69.6

          \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \]

   6. Applied rewrites 69.6%

      \[\leadsto 2 \cdot \left(x \cdot y - \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot \color{blue}{i}\right) \]

   if -1.99999999999999999e-55 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.9999999999999996e-82

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      3. lower-*.f64 55.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right) \]

   4. Applied rewrites 55.2%

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

Alternative 7: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+199}:\\ \;\;\;\;\mathsf{fma}\left(c, b, a\right) \cdot \left(\left(i \cdot c\right) \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot i\right) \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -1e+199)
     (* (fma c b a) (* (* i c) -2.0))
     (if (<= t_1 2e+78)
       (* 2.0 (fma t z (* x y)))
       (* (* -2.0 i) (* (fma c b a) c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_1 <= -1e+199) {
		tmp = fma(c, b, a) * ((i * c) * -2.0);
	} else if (t_1 <= 2e+78) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = (-2.0 * i) * (fma(c, b, a) * c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_1 <= -1e+199)
		tmp = Float64(fma(c, b, a) * Float64(Float64(i * c) * -2.0));
	elseif (t_1 <= 2e+78)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(Float64(-2.0 * i) * Float64(fma(c, b, a) * c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+199], N[(N[(c * b + a), $MachinePrecision] * N[(N[(i * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+78], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * i), $MachinePrecision] * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.0000000000000001e199

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      5. lower-*.f64 47.1

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(a + b \cdot c\right) \cdot \color{blue}{i}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(a + b \cdot c\right) \cdot i\right) \]

      7. +-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(b \cdot c + a\right) \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(b \cdot c + a\right) \cdot i\right) \]

      9. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right) \]

      10. associate-*r* N/A

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

      11. associate-*l* N/A

          \[\leadsto \left(-2 \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right) \cdot i \]

      12. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\right) \cdot i \]

      13. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\right) \cdot i \]

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \cdot -2 \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \cdot -2 \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot i\right)\right) \cdot -2 \]

      20. associate-*l* N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 48.6%

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

   if -1.0000000000000001e199 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000002e78

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      3. lower-*.f64 55.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right) \]

   4. Applied rewrites 55.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)} \]

   if 2.00000000000000002e78 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      5. lower-*.f64 47.1

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

   6. Applied rewrites 46.8%

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

Alternative 8: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(c, b, a\right) \cdot \left(\left(i \cdot c\right) \cdot -2\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+199}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma c b a) (* (* i c) -2.0))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -1e+199)
     t_1
     (if (<= t_2 2e+78) (* 2.0 (fma t z (* x y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(c, b, a) * ((i * c) * -2.0);
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -1e+199) {
		tmp = t_1;
	} else if (t_2 <= 2e+78) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(c, b, a) * Float64(Float64(i * c) * -2.0))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -1e+199)
		tmp = t_1;
	elseif (t_2 <= 2e+78)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * b + a), $MachinePrecision] * N[(N[(i * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+199], t$95$1, If[LessEqual[t$95$2, 2e+78], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.0000000000000001e199 or 2.00000000000000002e78 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      5. lower-*.f64 47.1

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(a + b \cdot c\right) \cdot \color{blue}{i}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(a + b \cdot c\right) \cdot i\right) \]

      7. +-commutative N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(b \cdot c + a\right) \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\left(b \cdot c + a\right) \cdot i\right) \]

      9. lift-fma.f64 N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right) \]

      10. associate-*r* N/A

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

      11. associate-*l* N/A

          \[\leadsto \left(-2 \cdot \left(c \cdot \mathsf{fma}\left(b, c, a\right)\right)\right) \cdot i \]

      12. *-commutative N/A

          \[\leadsto \left(-2 \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\right) \cdot i \]

      13. lift-*.f64 N/A

          \[\leadsto \left(-2 \cdot \left(\mathsf{fma}\left(b, c, a\right) \cdot c\right)\right) \cdot i \]

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \cdot -2 \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(b, c, a\right) \cdot c\right) \cdot i\right) \cdot -2 \]

      19. associate-*l* N/A

          \[\leadsto \left(\mathsf{fma}\left(b, c, a\right) \cdot \left(c \cdot i\right)\right) \cdot -2 \]

      20. associate-*l* N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 48.6%

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

   if -1.0000000000000001e199 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000002e78

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      3. lower-*.f64 55.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right) \]

   4. Applied rewrites 55.2%

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

Alternative 9: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot \left(-2 \cdot c\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (fma c b a) i) (* -2.0 c))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -5e+229)
     t_1
     (if (<= t_2 2e+78) (* 2.0 (fma t z (* x y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (fma(c, b, a) * i) * (-2.0 * c);
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -5e+229) {
		tmp = t_1;
	} else if (t_2 <= 2e+78) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(fma(c, b, a) * i) * Float64(-2.0 * c))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+229)
		tmp = t_1;
	elseif (t_2 <= 2e+78)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(c * b + a), $MachinePrecision] * i), $MachinePrecision] * N[(-2.0 * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+229], t$95$1, If[LessEqual[t$95$2, 2e+78], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000005e229 or 2.00000000000000002e78 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      5. lower-*.f64 47.1

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(-2 \cdot c\right) \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{\left(-2 \cdot c\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \color{blue}{\left(-2 \cdot c\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(i \cdot \left(a + b \cdot c\right)\right) \cdot \left(\color{blue}{-2} \cdot c\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\left(a + b \cdot c\right) \cdot i\right) \cdot \left(\color{blue}{-2} \cdot c\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(\left(a + b \cdot c\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      9. +-commutative N/A

         \[\leadsto \left(\left(b \cdot c + a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(b \cdot c + a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      12. lower-*.f64 N/A

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

      13. lift-fma.f64 N/A

          \[\leadsto \left(\left(b \cdot c + a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      14. *-commutative N/A

          \[\leadsto \left(\left(c \cdot b + a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(c, b, a\right) \cdot i\right) \cdot \left(-2 \cdot c\right) \]

      16. lower-*.f64 47.1

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

   6. Applied rewrites 47.1%

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

   if -5.0000000000000005e229 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000002e78

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      3. lower-*.f64 55.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right) \]

   4. Applied rewrites 55.2%

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

Alternative 10: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := -2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+277}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+307}:\\ \;\;\;\;-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* c (* b (* c i))))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -2e+277)
     t_1
     (if (<= t_2 5e+197)
       (* 2.0 (fma t z (* x y)))
       (if (<= t_2 1e+307) (* -2.0 (* a (* c i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -2.0 * (c * (b * (c * i)));
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -2e+277) {
		tmp = t_1;
	} else if (t_2 <= 5e+197) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else if (t_2 <= 1e+307) {
		tmp = -2.0 * (a * (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(-2.0 * Float64(c * Float64(b * Float64(c * i))))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+277)
		tmp = t_1;
	elseif (t_2 <= 5e+197)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	elseif (t_2 <= 1e+307)
		tmp = Float64(-2.0 * Float64(a * Float64(c * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(-2.0 * N[(c * N[(b * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+277], t$95$1, If[LessEqual[t$95$2, 5e+197], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+307], N[(-2.0 * N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.00000000000000001e277 or 9.99999999999999986e306 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \color{blue}{\left(i \cdot \left(a + b \cdot c\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \color{blue}{\left(a + b \cdot c\right)}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + \color{blue}{b \cdot c}\right)\right)\right) \]

      5. lower-*.f64 47.1

         \[\leadsto -2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot \color{blue}{c}\right)\right)\right) \]

   4. Applied rewrites 47.1%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto -2 \cdot \left(c \cdot \left(b \cdot \color{blue}{\left(c \cdot i\right)}\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot \color{blue}{i}\right)\right)\right) \]

      2. lower-*.f64 34.1

         \[\leadsto -2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right) \]

   7. Applied rewrites 34.1%

      \[\leadsto -2 \cdot \left(c \cdot \left(b \cdot \color{blue}{\left(c \cdot i\right)}\right)\right) \]

   if -2.00000000000000001e277 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000009e197

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      3. lower-*.f64 55.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right) \]

   4. Applied rewrites 55.2%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)} \]

   if 5.00000000000000009e197 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 9.99999999999999986e306

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 25.0

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

   4. Applied rewrites 25.0%

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

Alternative 11: 62.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := -2 \cdot \left(a \cdot \left(c \cdot i\right)\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+197}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* a (* c i)))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 5e+197) (* 2.0 (fma t z (* x y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -2.0 * (a * (c * i));
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+197) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(-2.0 * Float64(a * Float64(c * i)))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+197)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(-2.0 * N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+197], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 5.00000000000000009e197 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 25.0

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

   4. Applied rewrites 25.0%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5.00000000000000009e197

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z + x \cdot y\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z + x \cdot y\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, \color{blue}{z}, x \cdot y\right) \]

      3. lower-*.f64 55.2

         \[\leadsto 2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right) \]

   4. Applied rewrites 55.2%

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

Alternative 12: 44.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \left(y + y\right) \cdot x\\ \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-100}:\\ \;\;\;\;\left(t + t\right) \cdot z\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+77}:\\ \;\;\;\;-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ y y) x)))
   (if (<= (* x y) -4e+37)
     t_1
     (if (<= (* x y) -1e-100)
       (* (+ t t) z)
       (if (<= (* x y) 4e+77) (* -2.0 (* a (* c i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y + y) * x;
	double tmp;
	if ((x * y) <= -4e+37) {
		tmp = t_1;
	} else if ((x * y) <= -1e-100) {
		tmp = (t + t) * z;
	} else if ((x * y) <= 4e+77) {
		tmp = -2.0 * (a * (c * i));
	} else {
		tmp = t_1;
	}
	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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y + y) * x
    if ((x * y) <= (-4d+37)) then
        tmp = t_1
    else if ((x * y) <= (-1d-100)) then
        tmp = (t + t) * z
    else if ((x * y) <= 4d+77) then
        tmp = (-2.0d0) * (a * (c * i))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y + y) * x;
	double tmp;
	if ((x * y) <= -4e+37) {
		tmp = t_1;
	} else if ((x * y) <= -1e-100) {
		tmp = (t + t) * z;
	} else if ((x * y) <= 4e+77) {
		tmp = -2.0 * (a * (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (y + y) * x
	tmp = 0
	if (x * y) <= -4e+37:
		tmp = t_1
	elif (x * y) <= -1e-100:
		tmp = (t + t) * z
	elif (x * y) <= 4e+77:
		tmp = -2.0 * (a * (c * i))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y + y) * x)
	tmp = 0.0
	if (Float64(x * y) <= -4e+37)
		tmp = t_1;
	elseif (Float64(x * y) <= -1e-100)
		tmp = Float64(Float64(t + t) * z);
	elseif (Float64(x * y) <= 4e+77)
		tmp = Float64(-2.0 * Float64(a * Float64(c * i)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (y + y) * x;
	tmp = 0.0;
	if ((x * y) <= -4e+37)
		tmp = t_1;
	elseif ((x * y) <= -1e-100)
		tmp = (t + t) * z;
	elseif ((x * y) <= 4e+77)
		tmp = -2.0 * (a * (c * i));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y + y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -4e+37], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -1e-100], N[(N[(t + t), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4e+77], N[(-2.0 * N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.99999999999999982e37 or 3.99999999999999993e77 < (*.f64 x y)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot i\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(c \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      3. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \color{blue}{\left(a + b \cdot c\right)}\right) \cdot i\right) \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      5. distribute-lft-in N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(c \cdot \left(b \cdot c\right) + c \cdot a\right)} \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \color{blue}{\left(b \cdot c\right)} + c \cdot a\right) \cdot i\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \color{blue}{\left(c \cdot b\right)} + c \cdot a\right) \cdot i\right) \]

      8. associate-*r* N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\color{blue}{\left(c \cdot c\right) \cdot b} + c \cdot a\right) \cdot i\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(c \cdot c\right) \cdot b + \color{blue}{a \cdot c}\right) \cdot i\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\mathsf{fma}\left(c \cdot c, b, a \cdot c\right)} \cdot i\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(\color{blue}{c \cdot c}, b, a \cdot c\right) \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(c \cdot c, b, \color{blue}{c \cdot a}\right) \cdot i\right) \]

      13. lower-*.f64 83.5

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(c \cdot c, b, \color{blue}{c \cdot a}\right) \cdot i\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\mathsf{fma}\left(c \cdot c, b, c \cdot a\right)} \cdot i\right) \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y\right)} \]

      2. lower-*.f64 30.3

         \[\leadsto 2 \cdot \left(x \cdot \color{blue}{y}\right) \]

   6. Applied rewrites 30.3%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot \color{blue}{y}\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot \color{blue}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(2 \cdot y\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot y\right) \cdot \color{blue}{x} \]

      6. count-2-rev N/A

         \[\leadsto \left(y + y\right) \cdot x \]

      7. lower-+.f64 30.3

         \[\leadsto \left(y + y\right) \cdot x \]

   8. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\left(y + y\right) \cdot x} \]

   if -3.99999999999999982e37 < (*.f64 x y) < -1e-100

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

      2. lower-*.f64 28.2

         \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

   4. Applied rewrites 28.2%

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(2 \cdot t\right) \cdot \color{blue}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot t\right) \cdot \color{blue}{z} \]

      5. count-2-rev N/A

         \[\leadsto \left(t + t\right) \cdot z \]

      6. lower-+.f64 28.2

         \[\leadsto \left(t + t\right) \cdot z \]

   6. Applied rewrites 28.2%

      \[\leadsto \color{blue}{\left(t + t\right) \cdot z} \]

   if -1e-100 < (*.f64 x y) < 3.99999999999999993e77

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 25.0

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

   4. Applied rewrites 25.0%

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

Alternative 13: 41.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \left(t + t\right) \cdot z\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+107}:\\ \;\;\;\;\left(y + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ t t) z)))
   (if (<= (* z t) -2e+65) t_1 (if (<= (* z t) 4e+107) (* (+ y y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t + t) * z;
	double tmp;
	if ((z * t) <= -2e+65) {
		tmp = t_1;
	} else if ((z * t) <= 4e+107) {
		tmp = (y + y) * x;
	} else {
		tmp = t_1;
	}
	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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t + t) * z
    if ((z * t) <= (-2d+65)) then
        tmp = t_1
    else if ((z * t) <= 4d+107) then
        tmp = (y + y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t + t) * z;
	double tmp;
	if ((z * t) <= -2e+65) {
		tmp = t_1;
	} else if ((z * t) <= 4e+107) {
		tmp = (y + y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (t + t) * z
	tmp = 0
	if (z * t) <= -2e+65:
		tmp = t_1
	elif (z * t) <= 4e+107:
		tmp = (y + y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t + t) * z)
	tmp = 0.0
	if (Float64(z * t) <= -2e+65)
		tmp = t_1;
	elseif (Float64(z * t) <= 4e+107)
		tmp = Float64(Float64(y + y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (t + t) * z;
	tmp = 0.0;
	if ((z * t) <= -2e+65)
		tmp = t_1;
	elseif ((z * t) <= 4e+107)
		tmp = (y + y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(t + t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+65], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 4e+107], N[(N[(y + y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e65 or 3.9999999999999999e107 < (*.f64 z t)

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

      2. lower-*.f64 28.2

         \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

   4. Applied rewrites 28.2%

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(2 \cdot t\right) \cdot \color{blue}{z} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot t\right) \cdot \color{blue}{z} \]

      5. count-2-rev N/A

         \[\leadsto \left(t + t\right) \cdot z \]

      6. lower-+.f64 28.2

         \[\leadsto \left(t + t\right) \cdot z \]

   6. Applied rewrites 28.2%

      \[\leadsto \color{blue}{\left(t + t\right) \cdot z} \]

   if -2e65 < (*.f64 z t) < 3.9999999999999999e107

   1. Initial program 90.0%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(\left(a + b \cdot c\right) \cdot c\right)} \cdot i\right) \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(c \cdot \left(a + b \cdot c\right)\right)} \cdot i\right) \]

      3. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \color{blue}{\left(a + b \cdot c\right)}\right) \cdot i\right) \]

      4. +-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \color{blue}{\left(b \cdot c + a\right)}\right) \cdot i\right) \]

      5. distribute-lft-in N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\left(c \cdot \left(b \cdot c\right) + c \cdot a\right)} \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \color{blue}{\left(b \cdot c\right)} + c \cdot a\right) \cdot i\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(c \cdot \color{blue}{\left(c \cdot b\right)} + c \cdot a\right) \cdot i\right) \]

      8. associate-*r* N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\color{blue}{\left(c \cdot c\right) \cdot b} + c \cdot a\right) \cdot i\right) \]

      9. *-commutative N/A

         \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(c \cdot c\right) \cdot b + \color{blue}{a \cdot c}\right) \cdot i\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\mathsf{fma}\left(c \cdot c, b, a \cdot c\right)} \cdot i\right) \]

      11. lower-*.f64 N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(\color{blue}{c \cdot c}, b, a \cdot c\right) \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(c \cdot c, b, \color{blue}{c \cdot a}\right) \cdot i\right) \]

      13. lower-*.f64 83.5

          \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \mathsf{fma}\left(c \cdot c, b, \color{blue}{c \cdot a}\right) \cdot i\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \color{blue}{\mathsf{fma}\left(c \cdot c, b, c \cdot a\right)} \cdot i\right) \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y\right)} \]

      2. lower-*.f64 30.3

         \[\leadsto 2 \cdot \left(x \cdot \color{blue}{y}\right) \]

   6. Applied rewrites 30.3%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 2 \cdot \color{blue}{\left(x \cdot y\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(x \cdot \color{blue}{y}\right) \]

      3. *-commutative N/A

         \[\leadsto 2 \cdot \left(y \cdot \color{blue}{x}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(2 \cdot y\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot y\right) \cdot \color{blue}{x} \]

      6. count-2-rev N/A

         \[\leadsto \left(y + y\right) \cdot x \]

      7. lower-+.f64 30.3

         \[\leadsto \left(y + y\right) \cdot x \]

   8. Applied rewrites 30.3%

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

Alternative 14: 28.2% accurate, 4.0× speedup

Math

\[\left(t + t\right) \cdot z \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (* (+ t t) z))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (t + t) * z;
}

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(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (t + t) * z
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (t + t) * z;
}

Python

def code(x, y, z, t, a, b, c, i):
	return (t + t) * z

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(t + t) * z)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (t + t) * z;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(t + t), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

2. Taylor expanded in z around inf

   \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

   2. lower-*.f64 28.2

      \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

4. Applied rewrites 28.2%

   \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 2 \cdot \color{blue}{\left(t \cdot z\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \left(t \cdot \color{blue}{z}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(2 \cdot t\right) \cdot \color{blue}{z} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot t\right) \cdot \color{blue}{z} \]

   5. count-2-rev N/A

      \[\leadsto \left(t + t\right) \cdot z \]

   6. lower-+.f64 28.2

      \[\leadsto \left(t + t\right) \cdot z \]

6. Applied rewrites 28.2%

   \[\leadsto \color{blue}{\left(t + t\right) \cdot z} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025171 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))