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

Percentage Accurate: 90.2% → 95.4%

Time: 4.7s

Alternatives: 12

Speedup: 1.0×

• 47.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \end{array} \]

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: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \end{array} \]

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: 95.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (fma (fma c b a) (- (* i c)) (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), -(i * c), 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(i * c)), 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[(i * c), $MachinePrecision]) + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

   \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
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. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 94.9

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 95.4

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 95.4

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

3. Applied rewrites 95.4%

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

Alternative 2: 95.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -i \cdot c\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), t\_1, t \cdot z\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+232}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, t \cdot z - i \cdot \left(\mathsf{fma}\left(c, b, a\right) \cdot c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), t\_1, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (- (* i c))) (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 (- INFINITY))
     (* 2.0 (fma (fma c b a) t_1 (* t z)))
     (if (<= t_2 2e+232)
       (* 2.0 (fma y x (- (* t z) (* i (* (fma c b a) c)))))
       (* 2.0 (fma (fma c b a) t_1 (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = -(i * c);
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 2.0 * fma(fma(c, b, a), t_1, (t * z));
	} else if (t_2 <= 2e+232) {
		tmp = 2.0 * fma(y, x, ((t * z) - (i * (fma(c, b, a) * c))));
	} else {
		tmp = 2.0 * fma(fma(c, b, a), t_1, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(-Float64(i * c))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(2.0 * fma(fma(c, b, a), t_1, Float64(t * z)));
	elseif (t_2 <= 2e+232)
		tmp = Float64(2.0 * fma(y, x, Float64(Float64(t * z) - Float64(i * Float64(fma(c, b, a) * c)))));
	else
		tmp = Float64(2.0 * fma(fma(c, b, a), t_1, Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = (-N[(i * c), $MachinePrecision])}, Block[{t$95$2 = N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(2.0 * N[(N[(c * b + a), $MachinePrecision] * t$95$1 + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+232], 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[(c * b + a), $MachinePrecision] * t$95$1 + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 90.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 95.4

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 95.4

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

   3. Applied rewrites 95.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 72.1

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

   6. Applied rewrites 72.1%

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

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 2.00000000000000011e232

   1. Initial program 90.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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 91.8

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

      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 91.8

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

      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 91.8

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

      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 91.8

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

   3. Applied rewrites 91.8%

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

   if 2.00000000000000011e232 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 95.4

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 95.4

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

   3. Applied rewrites 95.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 89.0

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

   6. Applied rewrites 89.0%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 72.7%

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

Alternative 3: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(c, b, a\right), -i \cdot c, t \cdot z\right)\\ t_2 := \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* 2.0 (fma (fma c b a) (- (* i c)) (* t z))))
        (t_2 (* (* (+ a (* b c)) c) i)))
   (if (<= t_2 -5e+30)
     t_1
     (if (<= t_2 2e+74) (* 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 = 2.0 * fma(fma(c, b, a), -(i * c), (t * z));
	double t_2 = ((a + (b * c)) * c) * i;
	double tmp;
	if (t_2 <= -5e+30) {
		tmp = t_1;
	} else if (t_2 <= 2e+74) {
		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(2.0 * fma(fma(c, b, a), Float64(-Float64(i * c)), Float64(t * z)))
	t_2 = Float64(Float64(Float64(a + Float64(b * c)) * c) * i)
	tmp = 0.0
	if (t_2 <= -5e+30)
		tmp = t_1;
	elseif (t_2 <= 2e+74)
		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[(2.0 * N[(N[(c * b + a), $MachinePrecision] * (-N[(i * c), $MachinePrecision]) + N[(t * z), $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, -5e+30], t$95$1, If[LessEqual[t$95$2, 2e+74], 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) < -4.9999999999999998e30 or 1.9999999999999999e74 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 90.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 95.4

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 95.4

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

   3. Applied rewrites 95.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 72.1

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

   6. Applied rewrites 72.1%

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

   if -4.9999999999999998e30 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 1.9999999999999999e74

   1. Initial program 90.2%

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

   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 74.0

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

   4. Applied rewrites 74.0%

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

Alternative 4: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+87}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right) \cdot \left(-i\right), c, z \cdot t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(b, c, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ a (* b c)) c)))
   (if (<= t_1 -5e+87)
     (* 2.0 (fma (* (fma b c a) (- i)) c (* z t)))
     (if (<= t_1 5e+131)
       (* 2.0 (- (fma t z (* x y)) (* a (* c i))))
       (* (* (* -2.0 c) i) (fma b c a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a + (b * c)) * c;
	double tmp;
	if (t_1 <= -5e+87) {
		tmp = 2.0 * fma((fma(b, c, a) * -i), c, (z * t));
	} else if (t_1 <= 5e+131) {
		tmp = 2.0 * (fma(t, z, (x * y)) - (a * (c * i)));
	} else {
		tmp = ((-2.0 * c) * i) * fma(b, c, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_1 <= -5e+87)
		tmp = Float64(2.0 * fma(Float64(fma(b, c, a) * Float64(-i)), c, Float64(z * t)));
	elseif (t_1 <= 5e+131)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(x * y)) - Float64(a * Float64(c * i))));
	else
		tmp = Float64(Float64(Float64(-2.0 * c) * i) * fma(b, c, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+87], N[(2.0 * N[(N[(N[(b * c + a), $MachinePrecision] * (-i)), $MachinePrecision] * c + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+131], N[(2.0 * N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * c), $MachinePrecision] * i), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -4.9999999999999998e87

   1. Initial program 90.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 68.6

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

   4. Applied rewrites 68.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lift-neg.f64 N/A

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

      15. +-commutative N/A

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

   6. Applied rewrites 69.4%

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

   if -4.9999999999999998e87 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.99999999999999995e131

   1. Initial program 90.2%

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

   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 74.0

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

   4. Applied rewrites 74.0%

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

   if 4.99999999999999995e131 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.2%

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

   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. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 48.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 48.9

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

   6. Applied rewrites 48.9%

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

Alternative 5: 86.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+228}:\\ \;\;\;\;\left(z \cdot t - \left(\mathsf{fma}\left(b, c, a\right) \cdot i\right) \cdot c\right) \cdot 2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(b, c, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ a (* b c)) c)))
   (if (<= t_1 -5e+228)
     (* (- (* z t) (* (* (fma b c a) i) c)) 2.0)
     (if (<= t_1 5e+131)
       (* 2.0 (- (fma t z (* x y)) (* a (* c i))))
       (* (* (* -2.0 c) i) (fma b c a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a + (b * c)) * c;
	double tmp;
	if (t_1 <= -5e+228) {
		tmp = ((z * t) - ((fma(b, c, a) * i) * c)) * 2.0;
	} else if (t_1 <= 5e+131) {
		tmp = 2.0 * (fma(t, z, (x * y)) - (a * (c * i)));
	} else {
		tmp = ((-2.0 * c) * i) * fma(b, c, a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_1 <= -5e+228)
		tmp = Float64(Float64(Float64(z * t) - Float64(Float64(fma(b, c, a) * i) * c)) * 2.0);
	elseif (t_1 <= 5e+131)
		tmp = Float64(2.0 * Float64(fma(t, z, Float64(x * y)) - Float64(a * Float64(c * i))));
	else
		tmp = Float64(Float64(Float64(-2.0 * c) * i) * fma(b, c, a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+228], N[(N[(N[(z * t), $MachinePrecision] - N[(N[(N[(b * c + a), $MachinePrecision] * i), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[t$95$1, 5e+131], N[(2.0 * N[(N[(t * z + N[(x * y), $MachinePrecision]), $MachinePrecision] - N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-2.0 * c), $MachinePrecision] * i), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (+.f64 a (*.f64 b c)) c) < -5e228

   1. Initial program 90.2%

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]
   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. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 94.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 95.4

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 95.4

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

   3. Applied rewrites 95.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 72.1

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

   6. Applied rewrites 72.1%

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

   8. Applied rewrites 69.2%

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

   if -5e228 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.99999999999999995e131

   1. Initial program 90.2%

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

   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 74.0

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

   4. Applied rewrites 74.0%

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

   if 4.99999999999999995e131 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.2%

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

   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. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 48.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 48.9

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

   6. Applied rewrites 48.9%

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

Alternative 6: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(b, c, a\right)\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+228}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(t, z, x \cdot y\right) - a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* -2.0 c) i) (fma b c a))) (t_2 (* (+ a (* b c)) c)))
   (if (<= t_2 -5e+228)
     t_1
     (if (<= t_2 5e+131) (* 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 = ((-2.0 * c) * i) * fma(b, c, a);
	double t_2 = (a + (b * c)) * c;
	double tmp;
	if (t_2 <= -5e+228) {
		tmp = t_1;
	} else if (t_2 <= 5e+131) {
		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(Float64(Float64(-2.0 * c) * i) * fma(b, c, a))
	t_2 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -5e+228)
		tmp = t_1;
	elseif (t_2 <= 5e+131)
		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[(N[(-2.0 * c), $MachinePrecision] * i), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+228], t$95$1, If[LessEqual[t$95$2, 5e+131], 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 a (*.f64 b c)) c) < -5e228 or 4.99999999999999995e131 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.2%

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

   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. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 48.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 48.9

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

   6. Applied rewrites 48.9%

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

   if -5e228 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.99999999999999995e131

   1. Initial program 90.2%

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

   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 74.0

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

   4. Applied rewrites 74.0%

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

Alternative 7: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-2 \cdot c\right) \cdot i\right) \cdot \mathsf{fma}\left(b, c, a\right)\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* (* -2.0 c) i) (fma b c a))) (t_2 (* (+ a (* b c)) c)))
   (if (<= t_2 -5e+162)
     t_1
     (if (<= t_2 5e+131) (* 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) * i) * fma(b, c, a);
	double t_2 = (a + (b * c)) * c;
	double tmp;
	if (t_2 <= -5e+162) {
		tmp = t_1;
	} else if (t_2 <= 5e+131) {
		tmp = 2.0 * fma(t, z, (x * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(-2.0 * c) * i) * fma(b, c, a))
	t_2 = Float64(Float64(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -5e+162)
		tmp = t_1;
	elseif (t_2 <= 5e+131)
		tmp = Float64(2.0 * fma(t, z, Float64(x * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(-2.0 * c), $MachinePrecision] * i), $MachinePrecision] * N[(b * c + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+162], t$95$1, If[LessEqual[t$95$2, 5e+131], 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 a (*.f64 b c)) c) < -4.9999999999999997e162 or 4.99999999999999995e131 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.2%

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

   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. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 48.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 48.9

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

   6. Applied rewrites 48.9%

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

   if -4.9999999999999997e162 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.99999999999999995e131

   1. Initial program 90.2%

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

   2. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 55.2

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

   4. Applied rewrites 55.2%

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

Alternative 8: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(c \cdot \left(b \cdot \left(c \cdot i\right)\right)\right)\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+228}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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)))
   (if (<= t_2 -5e+228)
     t_1
     (if (<= t_2 5e+131) (* 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;
	double tmp;
	if (t_2 <= -5e+228) {
		tmp = t_1;
	} else if (t_2 <= 5e+131) {
		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(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -5e+228)
		tmp = t_1;
	elseif (t_2 <= 5e+131)
		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[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+228], t$95$1, If[LessEqual[t$95$2, 5e+131], 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 a (*.f64 b c)) c) < -5e228 or 4.99999999999999995e131 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.2%

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

   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 34.1

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

   7. Applied rewrites 34.1%

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

   if -5e228 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.99999999999999995e131

   1. Initial program 90.2%

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

   2. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 55.2

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

   4. Applied rewrites 55.2%

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

Alternative 9: 59.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -2 \cdot \left(a \cdot \left(c \cdot i\right)\right)\\ t_2 := \left(a + b \cdot c\right) \cdot c\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(t, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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)))
   (if (<= t_2 -5e+162)
     t_1
     (if (<= t_2 5e+131) (* 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;
	double tmp;
	if (t_2 <= -5e+162) {
		tmp = t_1;
	} else if (t_2 <= 5e+131) {
		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(a + Float64(b * c)) * c)
	tmp = 0.0
	if (t_2 <= -5e+162)
		tmp = t_1;
	elseif (t_2 <= 5e+131)
		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[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+162], t$95$1, If[LessEqual[t$95$2, 5e+131], 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 a (*.f64 b c)) c) < -4.9999999999999997e162 or 4.99999999999999995e131 < (*.f64 (+.f64 a (*.f64 b c)) c)

   1. Initial program 90.2%

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

   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 25.3

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

   4. Applied rewrites 25.3%

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

   if -4.9999999999999997e162 < (*.f64 (+.f64 a (*.f64 b c)) c) < 4.99999999999999995e131

   1. Initial program 90.2%

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

   2. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 55.2

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

   4. Applied rewrites 55.2%

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

Alternative 10: 44.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + t\right) \cdot z\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-315}:\\ \;\;\;\;\left(x + x\right) \cdot y\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+112}:\\ \;\;\;\;-2 \cdot \left(a \cdot \left(c \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ t t) z)))
   (if (<= (* z t) -2e+77)
     t_1
     (if (<= (* z t) -2e-315)
       (* (+ x x) y)
       (if (<= (* z t) 5e+112) (* -2.0 (* a (* c i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t + t) * z;
	double tmp;
	if ((z * t) <= -2e+77) {
		tmp = t_1;
	} else if ((z * t) <= -2e-315) {
		tmp = (x + x) * y;
	} else if ((z * t) <= 5e+112) {
		tmp = -2.0 * (a * (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t + t) * z
    if ((z * t) <= (-2d+77)) then
        tmp = t_1
    else if ((z * t) <= (-2d-315)) then
        tmp = (x + x) * y
    else if ((z * t) <= 5d+112) then
        tmp = (-2.0d0) * (a * (c * i))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t + t) * z;
	double tmp;
	if ((z * t) <= -2e+77) {
		tmp = t_1;
	} else if ((z * t) <= -2e-315) {
		tmp = (x + x) * y;
	} else if ((z * t) <= 5e+112) {
		tmp = -2.0 * (a * (c * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (t + t) * z
	tmp = 0
	if (z * t) <= -2e+77:
		tmp = t_1
	elif (z * t) <= -2e-315:
		tmp = (x + x) * y
	elif (z * t) <= 5e+112:
		tmp = -2.0 * (a * (c * i))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t + t) * z)
	tmp = 0.0
	if (Float64(z * t) <= -2e+77)
		tmp = t_1;
	elseif (Float64(z * t) <= -2e-315)
		tmp = Float64(Float64(x + x) * y);
	elseif (Float64(z * t) <= 5e+112)
		tmp = Float64(-2.0 * Float64(a * Float64(c * i)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (t + t) * z;
	tmp = 0.0;
	if ((z * t) <= -2e+77)
		tmp = t_1;
	elseif ((z * t) <= -2e-315)
		tmp = (x + x) * y;
	elseif ((z * t) <= 5e+112)
		tmp = -2.0 * (a * (c * i));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(t + t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+77], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -2e-315], N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+112], N[(-2.0 * N[(a * N[(c * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.99999999999999997e77 or 5e112 < (*.f64 z t)

   1. Initial program 90.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 28.7

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

   4. Applied rewrites 28.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 28.7

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

   6. Applied rewrites 28.7%

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

   if -1.99999999999999997e77 < (*.f64 z t) < -2.0000000019e-315

   1. Initial program 90.2%

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

   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 29.8

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

   4. Applied rewrites 29.8%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   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 29.8

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

   6. Applied rewrites 29.8%

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

   if -2.0000000019e-315 < (*.f64 z t) < 5e112

   1. Initial program 90.2%

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

   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 25.3

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

   4. Applied rewrites 25.3%

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

Alternative 11: 41.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + t\right) \cdot z\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\left(x + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (+ t t) z)))
   (if (<= (* z t) -2e+77) t_1 (if (<= (* z t) 5e+112) (* (+ x 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 = (t + t) * z;
	double tmp;
	if ((z * t) <= -2e+77) {
		tmp = t_1;
	} else if ((z * t) <= 5e+112) {
		tmp = (x + x) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t + t) * z
    if ((z * t) <= (-2d+77)) then
        tmp = t_1
    else if ((z * t) <= 5d+112) then
        tmp = (x + x) * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t + t) * z;
	double tmp;
	if ((z * t) <= -2e+77) {
		tmp = t_1;
	} else if ((z * t) <= 5e+112) {
		tmp = (x + x) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (t + t) * z
	tmp = 0
	if (z * t) <= -2e+77:
		tmp = t_1
	elif (z * t) <= 5e+112:
		tmp = (x + x) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t + t) * z)
	tmp = 0.0
	if (Float64(z * t) <= -2e+77)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e+112)
		tmp = Float64(Float64(x + x) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (t + t) * z;
	tmp = 0.0;
	if ((z * t) <= -2e+77)
		tmp = t_1;
	elseif ((z * t) <= 5e+112)
		tmp = (x + x) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(t + t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+77], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e+112], N[(N[(x + x), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999997e77 or 5e112 < (*.f64 z t)

   1. Initial program 90.2%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 28.7

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

   4. Applied rewrites 28.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 28.7

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

   6. Applied rewrites 28.7%

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

   if -1.99999999999999997e77 < (*.f64 z t) < 5e112

   1. Initial program 90.2%

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

   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 29.8

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

   4. Applied rewrites 29.8%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot y\right)} \]
   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 29.8

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

   6. Applied rewrites 29.8%

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

Alternative 12: 28.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(t + t\right) \cdot z \end{array} \]

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 90.2%

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

2. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 28.7

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

4. Applied rewrites 28.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f64 28.7

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

6. Applied rewrites 28.7%

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

Reproduce

herbie shell --seed 2025148 
(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))))