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

Percentage Accurate: 90.5% → 95.6%

Time: 4.9s

Alternatives: 15

Speedup: 1.0×

• 53.6% 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: 90.5% 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: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := a + b \cdot c\\ t_2 := t\_1 \cdot c\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-2 \cdot \left(c \cdot \mathsf{fma}\left(i \cdot b, c, i \cdot a\right)\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+212}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - i \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y - c \cdot \left(i \cdot t\_1\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (* b c))) (t_2 (* t_1 c)))
   (if (<= t_2 (- INFINITY))
     (* -2.0 (* c (fma (* i b) c (* i a))))
     (if (<= t_2 5e+212)
       (* 2.0 (fma y x (- (* t z) (* i (* (fma c b a) c)))))
       (* 2.0 (- (* x y) (* 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 = a + (b * c);
	double t_2 = t_1 * c;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -2.0 * (c * fma((i * b), c, (i * a)));
	} else if (t_2 <= 5e+212) {
		tmp = 2.0 * fma(y, x, ((t * z) - (i * (fma(c, b, a) * c))));
	} else {
		tmp = 2.0 * ((x * y) - (c * (i * t_1)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(b * c))
	t_2 = Float64(t_1 * c)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(c * fma(Float64(i * b), c, Float64(i * a))));
	elseif (t_2 <= 5e+212)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(i * Float64(fma(c, b, a) * c)))));
	else
		tmp = Float64(2.0 * Float64(Float64(x * y) - Float64(c * Float64(i * t_1))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * c), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-2.0 * N[(c * N[(N[(i * b), $MachinePrecision] * c + N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+212], N[(2.0 * N[(y * x + N[(N[(t * z), $MachinePrecision] - N[(i * N[(N[(c * b + a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x * y), $MachinePrecision] - N[(c * N[(i * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   if -inf.0 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.9999999999999999e212

   1. Initial program 90.5%

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

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 91.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.9%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.9%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 91.9%

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

   3. Applied rewrites 91.9%

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

   if 4.9999999999999999e212 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.5%

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

   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 69.8%

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

   4. Applied rewrites 69.8%

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

Alternative 2: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.5%

   \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
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 95.1%

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

   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 95.6%

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

   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 95.6%

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

3. Applied rewrites 95.6%

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

Alternative 3: 88.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := a + b \cdot c\\ t_2 := 2 \cdot \left(x \cdot y - c \cdot \left(i \cdot t\_1\right)\right)\\ t_3 := t\_1 \cdot c\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+82}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+212}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ a (* b c)))
        (t_2 (* 2.0 (- (* x y) (* c (* i t_1)))))
        (t_3 (* t_1 c)))
   (if (<= t_3 -2e+82)
     t_2
     (if (<= t_3 5e+212) (* 2.0 (fma y x (- (* t z) (* i (* a c))))) 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);
	double t_2 = 2.0 * ((x * y) - (c * (i * t_1)));
	double t_3 = t_1 * c;
	double tmp;
	if (t_3 <= -2e+82) {
		tmp = t_2;
	} else if (t_3 <= 5e+212) {
		tmp = 2.0 * fma(y, x, ((t * z) - (i * (a * c))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(a + Float64(b * c))
	t_2 = Float64(2.0 * Float64(Float64(x * y) - Float64(c * Float64(i * t_1))))
	t_3 = Float64(t_1 * c)
	tmp = 0.0
	if (t_3 <= -2e+82)
		tmp = t_2;
	elseif (t_3 <= 5e+212)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(i * Float64(a * c)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(x * y), $MachinePrecision] - N[(c * N[(i * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * c), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+82], t$95$2, If[LessEqual[t$95$3, 5e+212], N[(2.0 * N[(y * x + N[(N[(t * z), $MachinePrecision] - N[(i * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -1.9999999999999999e82 or 4.9999999999999999e212 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.5%

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

   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 69.8%

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

   4. Applied rewrites 69.8%

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

   if -1.9999999999999999e82 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.9999999999999999e212

   1. Initial program 90.5%

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

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 91.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.9%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.9%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 91.9%

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

   3. Applied rewrites 91.9%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 73.0%

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

   6. Applied rewrites 73.0%

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

Alternative 4: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(b, c, a\right) \cdot \left(\left(i \cdot c\right) \cdot -2\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+159}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - i \cdot \left(a \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma b c a) (* (* i c) -2.0))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -2e+212)
     t_1
     (if (<= t_2 2e+159) (* 2.0 (fma y x (- (* t z) (* i (* a c))))) 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(b, c, a) * ((i * c) * -2.0);
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -2e+212) {
		tmp = t_1;
	} else if (t_2 <= 2e+159) {
		tmp = 2.0 * fma(y, x, ((t * z) - (i * (a * c))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(b, c, a) * Float64(Float64(i * c) * -2.0))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+212)
		tmp = t_1;
	elseif (t_2 <= 2e+159)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(i * Float64(a * c)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(b * c + 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, -2e+212], t$95$1, If[LessEqual[t$95$2, 2e+159], N[(2.0 * N[(y * x + N[(N[(t * z), $MachinePrecision] - N[(i * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.9999999999999998e212 or 1.9999999999999999e159 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   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. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

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

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

      22. distribute-rgt-neg-out N/A

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

      23. lift-neg.f64 N/A

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

   6. Applied rewrites 48.8%

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

   if -1.9999999999999998e212 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999999e159

   1. Initial program 90.5%

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

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 91.9%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.9%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.9%

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 91.9%

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

   3. Applied rewrites 91.9%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 73.0%

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

   6. Applied rewrites 73.0%

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

Alternative 5: 87.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(b, c, a\right) \cdot \left(\left(i \cdot c\right) \cdot -2\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+159}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (fma b c a) (* (* i c) -2.0))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -2e+212)
     t_1
     (if (<= t_2 2e+159) (* 2.0 (- (fma t z (* x y)) (* 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 = fma(b, c, a) * ((i * c) * -2.0);
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -2e+212) {
		tmp = t_1;
	} else if (t_2 <= 2e+159) {
		tmp = 2.0 * (fma(t, z, (x * y)) - (a * (c * i)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(fma(b, c, a) * Float64(Float64(i * c) * -2.0))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+212)
		tmp = t_1;
	elseif (t_2 <= 2e+159)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(x * y)) - 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[(N[(b * c + 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, -2e+212], t$95$1, If[LessEqual[t$95$2, 2e+159], N[(2.0 * N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1.9999999999999998e212 or 1.9999999999999999e159 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   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. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

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

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

      22. distribute-rgt-neg-out N/A

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

      23. lift-neg.f64 N/A

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

   6. Applied rewrites 48.8%

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

   if -1.9999999999999998e212 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999999e159

   1. Initial program 90.5%

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

   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 74.9%

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

   4. Applied rewrites 74.9%

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

Alternative 6: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+209}:\\ \;\;\;\;\mathsf{fma}\left(b, c, a\right) \cdot \left(\left(i \cdot c\right) \cdot -2\right)\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(z + z, t, \left(y + y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot \left(-2 \cdot c\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -2e+209)
     (* (fma b c a) (* (* i c) -2.0))
     (if (<= t_1 2000.0)
       (fma (+ z z) t (* (+ y y) x))
       (* (* i (fma b c a)) (* -2.0 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 <= -2e+209) {
		tmp = fma(b, c, a) * ((i * c) * -2.0);
	} else if (t_1 <= 2000.0) {
		tmp = fma((z + z), t, ((y + y) * x));
	} else {
		tmp = (i * fma(b, c, a)) * (-2.0 * 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 <= -2e+209)
		tmp = Float64(fma(b, c, a) * Float64(Float64(i * c) * -2.0));
	elseif (t_1 <= 2000.0)
		tmp = fma(Float64(z + z), t, Float64(Float64(y + y) * x));
	else
		tmp = Float64(Float64(i * fma(b, c, a)) * Float64(-2.0 * 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, -2e+209], N[(N[(b * c + a), $MachinePrecision] * N[(N[(i * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(N[(z + z), $MachinePrecision] * t + N[(N[(y + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(b * c + a), $MachinePrecision]), $MachinePrecision] * N[(-2.0 * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000001e209

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   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. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

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

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

      22. distribute-rgt-neg-out N/A

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

      23. lift-neg.f64 N/A

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

   6. Applied rewrites 48.8%

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

   if -2.0000000000000001e209 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e3

   1. Initial program 90.5%

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

   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 55.6%

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

   4. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. count-2 N/A

         \[\leadsto 2 \cdot \left(t \cdot z\right) + \left(y + y\right) \cdot x \]

      8. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z\right) + \left(y + y\right) \cdot x \]

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(z \cdot t\right) + x \cdot \left(y + y\right) \]

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 55.5%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 55.5%

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

   6. Applied rewrites 55.5%

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

   if 2e3 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

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

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 47.4%

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

   6. Applied rewrites 47.4%

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

Alternative 7: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \left(i \cdot \mathsf{fma}\left(b, c, a\right)\right) \cdot \left(-2 \cdot c\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+209}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(z + z, t, \left(y + y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* i (fma b c a)) (* -2.0 c))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -2e+209)
     t_1
     (if (<= t_2 2000.0) (fma (+ z z) t (* (+ 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 = (i * fma(b, c, a)) * (-2.0 * c);
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -2e+209) {
		tmp = t_1;
	} else if (t_2 <= 2000.0) {
		tmp = fma((z + z), t, ((y + y) * x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(i * fma(b, c, a)) * Float64(-2.0 * c))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -2e+209)
		tmp = t_1;
	elseif (t_2 <= 2000.0)
		tmp = fma(Float64(z + z), t, Float64(Float64(y + y) * x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(i * N[(b * c + a), $MachinePrecision]), $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, -2e+209], t$95$1, If[LessEqual[t$95$2, 2000.0], N[(N[(z + z), $MachinePrecision] * t + N[(N[(y + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -2.0000000000000001e209 or 2e3 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

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

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 47.4%

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

   6. Applied rewrites 47.4%

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

   if -2.0000000000000001e209 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2e3

   1. Initial program 90.5%

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

   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 55.6%

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

   4. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. count-2 N/A

         \[\leadsto 2 \cdot \left(t \cdot z\right) + \left(y + y\right) \cdot x \]

      8. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z\right) + \left(y + y\right) \cdot x \]

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(z \cdot t\right) + x \cdot \left(y + y\right) \]

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 55.5%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 55.5%

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

   6. Applied rewrites 55.5%

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

Alternative 8: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := -2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+177}:\\ \;\;\;\;-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+260}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* c (* b (* c i))))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e+177)
       (* -2.0 (* a (* c i)))
       (if (<= t_2 1e+260) (* 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 * (c * (b * (c * i)));
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e+177) {
		tmp = -2.0 * (a * (c * i));
	} else if (t_2 <= 1e+260) {
		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(c * Float64(b * 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 <= -1e+177)
		tmp = Float64(-2.0 * Float64(a * Float64(c * i)));
	elseif (t_2 <= 1e+260)
		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[(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, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e+177], N[(-2.0 * N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+260], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0 or 1.0000000000000001e260 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

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

   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 33.6%

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

   7. Applied rewrites 33.6%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -1e177

   1. Initial program 90.5%

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

   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 25.9%

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

   4. Applied rewrites 25.9%

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

   if -1e177 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.0000000000000001e260

   1. Initial program 90.5%

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

   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 55.6%

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

   4. Applied rewrites 55.6%

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

Alternative 9: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+239}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(\left(b \cdot i\right) \cdot c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+260}:\\ \;\;\;\;\mathsf{fma}\left(z + z, t, \left(y + y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c\right) \cdot \left(\left(i \cdot c\right) \cdot -2\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -5e+239)
     (* -2.0 (* c (* (* b i) c)))
     (if (<= t_1 1e+260)
       (fma (+ z z) t (* (+ y y) x))
       (* (* b c) (* (* i c) -2.0))))))

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+239) {
		tmp = -2.0 * (c * ((b * i) * c));
	} else if (t_1 <= 1e+260) {
		tmp = fma((z + z), t, ((y + y) * x));
	} else {
		tmp = (b * c) * ((i * c) * -2.0);
	}
	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+239)
		tmp = Float64(-2.0 * Float64(c * Float64(Float64(b * i) * c)));
	elseif (t_1 <= 1e+260)
		tmp = fma(Float64(z + z), t, Float64(Float64(y + y) * x));
	else
		tmp = Float64(Float64(b * c) * Float64(Float64(i * c) * -2.0));
	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+239], N[(-2.0 * N[(c * N[(N[(b * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+260], N[(N[(z + z), $MachinePrecision] * t + N[(N[(y + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(b * c), $MachinePrecision] * N[(N[(i * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000001e239

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

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

   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 33.6%

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

   7. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 32.8%

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

   9. Applied rewrites 32.8%

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

   if -5.0000000000000001e239 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.0000000000000001e260

   1. Initial program 90.5%

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

   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 55.6%

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

   4. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. count-2 N/A

         \[\leadsto 2 \cdot \left(t \cdot z\right) + \left(y + y\right) \cdot x \]

      8. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(t \cdot z\right) + \left(y + y\right) \cdot x \]

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

          \[\leadsto 2 \cdot \left(z \cdot t\right) + x \cdot \left(y + y\right) \]

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 55.5%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 55.5%

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

   6. Applied rewrites 55.5%

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

   if 1.0000000000000001e260 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   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. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

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

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

      22. distribute-rgt-neg-out N/A

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

      23. lift-neg.f64 N/A

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

   6. Applied rewrites 48.8%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 33.5%

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

   9. Applied rewrites 33.5%

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

Alternative 10: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+239}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(\left(b \cdot i\right) \cdot c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+260}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c\right) \cdot \left(\left(i \cdot c\right) \cdot -2\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -5e+239)
     (* -2.0 (* c (* (* b i) c)))
     (if (<= t_1 1e+260)
       (* 2.0 (fma t z (* x y)))
       (* (* b c) (* (* i c) -2.0))))))

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+239) {
		tmp = -2.0 * (c * ((b * i) * c));
	} else if (t_1 <= 1e+260) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = (b * c) * ((i * c) * -2.0);
	}
	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+239)
		tmp = Float64(-2.0 * Float64(c * Float64(Float64(b * i) * c)));
	elseif (t_1 <= 1e+260)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(Float64(b * c) * Float64(Float64(i * c) * -2.0));
	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+239], N[(-2.0 * N[(c * N[(N[(b * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+260], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * c), $MachinePrecision] * N[(N[(i * c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000001e239

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

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

   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 33.6%

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

   7. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 32.8%

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

   9. Applied rewrites 32.8%

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

   if -5.0000000000000001e239 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.0000000000000001e260

   1. Initial program 90.5%

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

   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 55.6%

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

   4. Applied rewrites 55.6%

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

   if 1.0000000000000001e260 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{-2 \cdot \left(c \cdot \left(i \cdot \left(a + b \cdot c\right)\right)\right)} \]
   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. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

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

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

      22. distribute-rgt-neg-out N/A

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

      23. lift-neg.f64 N/A

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

   6. Applied rewrites 48.8%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 33.5%

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

   9. Applied rewrites 33.5%

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

Alternative 11: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+239}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(\left(b \cdot i\right) \cdot c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+260}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (+ a (* b c)) c) i)))
   (if (<= t_1 -5e+239)
     (* -2.0 (* c (* (* b i) c)))
     (if (<= t_1 1e+260)
       (* 2.0 (fma t z (* x y)))
       (* -2.0 (* c (* b (* c i))))))))

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+239) {
		tmp = -2.0 * (c * ((b * i) * c));
	} else if (t_1 <= 1e+260) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = -2.0 * (c * (b * (c * i)));
	}
	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+239)
		tmp = Float64(-2.0 * Float64(c * Float64(Float64(b * i) * c)));
	elseif (t_1 <= 1e+260)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = Float64(-2.0 * Float64(c * Float64(b * Float64(c * i))));
	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+239], N[(-2.0 * N[(c * N[(N[(b * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+260], N[(2.0 * N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(c * N[(b * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -5.0000000000000001e239

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

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

   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 33.6%

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

   7. Applied rewrites 33.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 32.8%

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

   9. Applied rewrites 32.8%

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

   if -5.0000000000000001e239 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.0000000000000001e260

   1. Initial program 90.5%

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

   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 55.6%

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

   4. Applied rewrites 55.6%

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

   if 1.0000000000000001e260 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.5%

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

   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 47.4%

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

   4. Applied rewrites 47.4%

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

   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 33.6%

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

   7. Applied rewrites 33.6%

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

Alternative 12: 63.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := -2 \cdot \left(a \cdot \left(c \cdot i\right)\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+284}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

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 -1e+177)
     t_1
     (if (<= t_2 2e+284) (* 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 <= -1e+177) {
		tmp = t_1;
	} else if (t_2 <= 2e+284) {
		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 <= -1e+177)
		tmp = t_1;
	elseif (t_2 <= 2e+284)
		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, -1e+177], t$95$1, If[LessEqual[t$95$2, 2e+284], 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) < -1e177 or 2.0000000000000002e284 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.5%

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

   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 25.9%

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

   4. Applied rewrites 25.9%

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

   if -1e177 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.0000000000000002e284

   1. Initial program 90.5%

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

   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 55.6%

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

   4. Applied rewrites 55.6%

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

Alternative 13: 44.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := -2 \cdot \left(a \cdot \left(c \cdot i\right)\right)\\ t_2 := \left(x + x\right) \cdot y\\ \mathbf{if}\;x \cdot y \leq -8 \cdot 10^{+76}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \cdot y \leq -1.2 \cdot 10^{-205}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(t + t\right) \cdot z\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* -2.0 (* a (* c i)))) (t_2 (* (+ x x) y)))
   (if (<= (* x y) -8e+76)
     t_2
     (if (<= (* x y) -1.2e-205)
       t_1
       (if (<= (* x y) 2e-94)
         (* (+ t t) z)
         (if (<= (* x y) 2e-26) t_1 t_2))))))

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 = (x + x) * y;
	double tmp;
	if ((x * y) <= -8e+76) {
		tmp = t_2;
	} else if ((x * y) <= -1.2e-205) {
		tmp = t_1;
	} else if ((x * y) <= 2e-94) {
		tmp = (t + t) * z;
	} else if ((x * y) <= 2e-26) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (-2.0d0) * (a * (c * i))
    t_2 = (x + x) * y
    if ((x * y) <= (-8d+76)) then
        tmp = t_2
    else if ((x * y) <= (-1.2d-205)) then
        tmp = t_1
    else if ((x * y) <= 2d-94) then
        tmp = (t + t) * z
    else if ((x * y) <= 2d-26) then
        tmp = t_1
    else
        tmp = t_2
    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 = -2.0 * (a * (c * i));
	double t_2 = (x + x) * y;
	double tmp;
	if ((x * y) <= -8e+76) {
		tmp = t_2;
	} else if ((x * y) <= -1.2e-205) {
		tmp = t_1;
	} else if ((x * y) <= 2e-94) {
		tmp = (t + t) * z;
	} else if ((x * y) <= 2e-26) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = -2.0 * (a * (c * i))
	t_2 = (x + x) * y
	tmp = 0
	if (x * y) <= -8e+76:
		tmp = t_2
	elif (x * y) <= -1.2e-205:
		tmp = t_1
	elif (x * y) <= 2e-94:
		tmp = (t + t) * z
	elif (x * y) <= 2e-26:
		tmp = t_1
	else:
		tmp = t_2
	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(x + x) * y)
	tmp = 0.0
	if (Float64(x * y) <= -8e+76)
		tmp = t_2;
	elseif (Float64(x * y) <= -1.2e-205)
		tmp = t_1;
	elseif (Float64(x * y) <= 2e-94)
		tmp = Float64(Float64(t + t) * z);
	elseif (Float64(x * y) <= 2e-26)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = -2.0 * (a * (c * i));
	t_2 = (x + x) * y;
	tmp = 0.0;
	if ((x * y) <= -8e+76)
		tmp = t_2;
	elseif ((x * y) <= -1.2e-205)
		tmp = t_1;
	elseif ((x * y) <= 2e-94)
		tmp = (t + t) * z;
	elseif ((x * y) <= 2e-26)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = 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[(x + x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -8e+76], t$95$2, If[LessEqual[N[(x * y), $MachinePrecision], -1.2e-205], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 2e-94], N[(N[(t + t), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-26], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -8.0000000000000004e76 or 2.0000000000000001e-26 < (*.f64 x y)

   1. Initial program 90.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 29.6%

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

   4. Applied rewrites 29.6%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   5. 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. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 29.6%

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

   6. Applied rewrites 29.6%

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

   if -8.0000000000000004e76 < (*.f64 x y) < -1.2000000000000001e-205 or 1.9999999999999999e-94 < (*.f64 x y) < 2.0000000000000001e-26

   1. Initial program 90.5%

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

   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 25.9%

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

   4. Applied rewrites 25.9%

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

   if -1.2000000000000001e-205 < (*.f64 x y) < 1.9999999999999999e-94

   1. Initial program 90.5%

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

   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 29.2%

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

   4. Applied rewrites 29.2%

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
   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 29.2%

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

   6. Applied rewrites 29.2%

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

Alternative 14: 42.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \left(x + x\right) \cdot y\\ \mathbf{if}\;x \cdot y \leq -8 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{-33}:\\ \;\;\;\;\left(t + t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ x x) y)))
   (if (<= (* x y) -8e+76) t_1 (if (<= (* x y) 1e-33) (* (+ t t) z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (x + x) * y;
	double tmp;
	if ((x * y) <= -8e+76) {
		tmp = t_1;
	} else if ((x * y) <= 1e-33) {
		tmp = (t + t) * z;
	} 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 = (x + x) * y
    if ((x * y) <= (-8d+76)) then
        tmp = t_1
    else if ((x * y) <= 1d-33) then
        tmp = (t + t) * z
    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 = (x + x) * y;
	double tmp;
	if ((x * y) <= -8e+76) {
		tmp = t_1;
	} else if ((x * y) <= 1e-33) {
		tmp = (t + t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (x + x) * y
	tmp = 0
	if (x * y) <= -8e+76:
		tmp = t_1
	elif (x * y) <= 1e-33:
		tmp = (t + t) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(x + x) * y)
	tmp = 0.0
	if (Float64(x * y) <= -8e+76)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e-33)
		tmp = Float64(Float64(t + t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (x + x) * y;
	tmp = 0.0;
	if ((x * y) <= -8e+76)
		tmp = t_1;
	elseif ((x * y) <= 1e-33)
		tmp = (t + t) * z;
	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[(x + x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -8e+76], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e-33], N[(N[(t + t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -8.0000000000000004e76 or 1.0000000000000001e-33 < (*.f64 x y)

   1. Initial program 90.5%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 29.6%

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

   4. Applied rewrites 29.6%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   5. 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. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 29.6%

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

   6. Applied rewrites 29.6%

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

   if -8.0000000000000004e76 < (*.f64 x y) < 1.0000000000000001e-33

   1. Initial program 90.5%

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

   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 29.2%

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

   4. Applied rewrites 29.2%

      \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
   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 29.2%

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

   6. Applied rewrites 29.2%

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

Alternative 15: 29.2% accurate, 4.1× 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 90.5%

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

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 29.2%

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

4. Applied rewrites 29.2%

   \[\leadsto \color{blue}{2 \cdot \left(t \cdot z\right)} \]
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 29.2%

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

6. Applied rewrites 29.2%

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

Reproduce

herbie shell --seed 2025205 
(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))))